Drogba's goal cancelled out James Vaughan's shock opener in the 49th minute but the Ivorian's leveller was shrouded in controversy.
Everton manager David Moyes was furious that referee Mark Halsey failed to punish Paulo Ferreira for a challenge on Mikel Arteta in the build-up to Chelsea's goal.
Moyes raced onto the pitch after Drogba had finished off a cross from Shaun Wright-Phillips and after being recalled by the fourth official, Halsey sent the Everton boss to the stands.
• Jose happy with record
Chelsea's unbeaten run pre-dates coach Jose Mourinho's arrival in the summer of 2004 with Arsenal the last team to defeat the Blues on home soil in February of that year.
But they have now matched Liverpool's glorious run at Anfield which was set between 1978 and 1981.
Everton, with a place in Europe next season already assured, were not expected to push the FA Cup finalists but Lee Carsley sent an early volley into the arms of Petr Cech inside the opening minute.
But Chelsea's Jon Obi Mikel was much more of a threat three minutes later when his right-foot shot from 25-yard brought a spectacular save from Tim Howard.
The Everton goalkeeper was forced to tip his effort over the crossbar as the ball headed for the top corner.
Cech was much more comfortable with a dipping volley from Vaughan but Everton demonstrated plenty of attacking verve in the opening quarter.
But Drogba almost added to his 31-goal tally when he sent a right-foot effort just over the bar in the 14th minute.
It was Salomon Kalou's turn to test Howard moments later when his mazy run into the penalty area was finished off with a low drive which forced the Everton keeper to dive low to his right to keep out.
In the 28th minute Joe Cole almost broke the deadlock after collecting the ball from Kalou's run down the left-flank.
But the England midfielder's low drive was deflected beyond Howard's right-hand upright much to the relief of the Everton goalkeeper.
Drogba was beginning to get the upper hand in his battle with Alan Stubbs and but after getting clear of the Everton defender once more in the 31st minute, he failed to hit the target with an angled drive which flashed past the post.
Chelsea were unable to make the most of their first-half dominance and on numerous occasions they failed to supply a telling final ball when it mattered.
Ferreira and Wright-Phillips were both guilty of failing to supply a decent cross into the penalty area and consequently Everton dealt with the threat easily.
Kalou, who had been a willing trier during the first half, was upended by Phil Neville when he raced towards the edge of the penalty area three minutes before the break.
The foul was committed in a perfect place for Drogba to try an open the scoring with a 22-yard free-kick but the Ivorian's effort missed the target.
Everton stunned the deposed champions by taking the lead in the 49th minute.
Mikel failed to stop the advancing Leon Osman in midfield and the Everton player ran on to provide Vaughan with a perfect pass.
The Everton striker found himself unmarked and with plenty of time to slide the ball under Cech to put Chelsea's dream of equalling Liverpool's 63-match unbeaten home record in real jeopardy.
They were almost caught out a second time three minutes later when Boulahrouz was again found wanting in defence.
This time Arteta got free on the right and his cross was volleyed towards the corner by the acrobatic Vaughan. However, Cech came to Chelsea's rescue by punching the ball to safety.
But Chelsea equalised in controversial circumstances in the 57th minute.
Everton manager David Moyes was furious Ferreira's challenge on Arteta went unpunished and it was compounded seconds later when Wright-Phillips crossed for Drogba to level with his 20th Premiership goal of the campaign.
Moyes ran on to the pitch to confront Halsey and after being recalled by the fourth official, was then sent to the stands for his protest.
But Everton still posed a real threat and Joseph Yobo was unlucky when he failed to direct his header on target in the 65th minute.
The Everton defender took advantage of Cech's hesitancy but his effort went over the bar.
But worryingly for Chelsea, they lost Mikel in the 74th minute with an injury which could jeopardise the Nigerian's chances of playing in the FA Cup final against Manchester United next Saturday.
Jose happy with record
Jose Mourinho was delighted to have equalled Liverpool's 63-match unbeaten
home record but claimed he might have just 15 fit players for next Saturday's FA
Cup final against Manchester United at the new Wembley.
The Chelsea coach spearheaded a lap of honour around the Stamford Bridge pitch
with his players after the game and insisted that it was a thank you gesture to
the fans rather than a fond farewell.
Didier Drogba's 32nd goal of the season cancelled out James Vaughan's opener
but Mourinho was already concerned with an injury list that threatens to scupper
his dreams of lifting the FA Cup - the only domestic trophy he has yet to win.
Midfielder Mikel John Obi limped off late in the second half and Mourinho has
added him to an injury list that already includes Andriy Shevchenko, Michael
Ballack, Ricardo Carvalho, Ashley Cole and Arjen Robben.
Mourinho declared: 'We have lost Mikel and the reality is that for a final
like we have on Saturday, I don't think we have 16 players.
'Mikel has a muscle injury so I don't believe he can play the game and in
such a game, where everybody wants to play, and we want to present the best
possible team, I don't think we have 16 players.
'Ballack is limping, Carvalho is limping, Robben limping, so the picture is
not a good one. It is a big final and something we want very much to win and the
11 players I play will play with a big spirit.
'Robben has a chance of being on the bench. He has not trained for months. He
trained for two days to be on the bench against Liverpool in the Champions
League at Anfield and to try and give us something, but has not trained since. I
don't think he can give us more than a little bit.
'Carvalho has no chance. Ashley Cole has a chance. Ballack and Shevchenko are
out and at this moment I have 15 players and two extra goalkeepers.
'I could not rest more players because I didn't have them. I gave a rest to
Claude Makelele and Michael Essien, all the other players were not ready.
'I could not protect more players. I was trying to end the season in the best
possible way - without a defeat and having the record.
'The positive thing is the record because we can say it has been three
consecutive years without defeat at home in the Premiership.
'We can say that we share the record with a historic Liverpool team. We are
very proud of that. We have the chance next season by not losing the first game
at home to have the record just for us.
'But to share it with such a Liverpool team is already great but to have it
to ourselves, I think it will be fantastic. We will become a team with a stamp
in English football. We have that chance.
'We finished the Premiership with some pride and honour. The last game we
lost in the league was in January. So we fought for four months without losing
another game in many difficult circumstances.
'But for winners the feeling is not good because we don't win. I am happy we
are not happy. It shows our nature.'
Mourinho and his two children led the players on a lap of honour after the
game and Mourinho insisted he would still be in charge when they try to beat
Liverpool's record in their first home game of next season.
When asked that if he thought he would leading the club into the next
campaign, he replied: 'Yes. I was just saying thank you to the fans for
everything.
'I was saying thank you for two fantastic seasons with some historical things
and the third season, not the same in terms of results, but in terms of giving
everything and working to the maximum of my ability to help my team and club to
win things, I think we did a good job.
'I did everything I could and gave everything. But I am not happy we didn't
win the Premiership. But I didn't lose motivation. I did the best I could.
'I don't feel happy - and I am happy that I am not happy. Next season we will
try again to win the Premiership but we know how difficult it is.'
Everton finished sixth in the Barclays Premiership with a 1-1 draw against
Chelsea despite having manager David Moyes banished to the stands for much of
the second half.
Moyes was furious that referee Mark Halsey failed to punish Chelsea defender
Paulo Ferreira for a challenge on Mikel Arteta in the build-up to Didier
Drogba's 57th minute equaliser.
Everton had gone ahead three minutes after the break through James Vaughan but
Drogba's leveller had Moyes so angry that he marched on to the pitch to
protest.
He was recalled to the touchline by the fourth official but Halsey then
ordered the Everton manager into the stands.
But Moyes, who took a spare seat right behind the away dugout, declared: 'I
think the decision was wrong and the players deserved my support today.
'I was sent off because I went on to the pitch. I went on and should not have
done. But where I came from you stand up for what it is right.
'Today, me going onto the pitch was right because the decision was wrong.
'It was a sell-out crowd but I noticed the only seat in the ground that was
empty.
'I had to apologise to a couple of parents because I think it was in the
family section.'
Everton were more than good value for their draw and but for a stunning save
from Chelsea's Petr Cech to deny Vaughan a second goal when the visitors led
1-0, the Toffees would have come away with all three points.
Moyes added: 'I thought the difference in the game was Petr Cech's save to
stop us making it 2-0.
'I thought we played terrifically well. I read Jose's (Mourinho) programme
notes where, as a manager, he had only lost one home game at any club he has
been at. That is unbelievable and really today I thought we were going to be the
second team.
'I thought our performance was excellent. I am sitting here disappointed
because I wanted to finish fifth. We have got European football but I thought we
had done enough to get a goal in front and thought maybe it was going to be our
day.
'We are a club who don't pay big wages and don't spend out outrageous amounts
on transfer fees.
'We have a great board, great players, and I've a good staff around me. We
would have taken sixth at the start of the season but with 30 minutes to go
today, we were sitting fifth.'
Chelsea 1 - 1 Everton
