Chelsea equalled Liverpool's top flight record of 63 home league games unbeaten, but Everton earned a deserved point from an uninspiring contest.
Drogba finished as the Premiership's leading goalscorer
The first half was woeful, Joe Cole missing the best chance as his left-foot shot trickled inches wide.
Things improved after the break and Everton took a shock lead, Leon Osman squaring for James Vaughan to slot in.
Didier Drogba levelled by sliding in Shaun Wright-Phillips' cross despite protests from the Everton bench.
Boss David Moyes felt that Paulo Ferreira had fouled Mikel Arteta in the build-up and he was sent to the stands for remonstrating with the officials.
But Moyes can be proud of his side's display and can now look forward to Everton's Uefa Cup campaign next term after the result secured a sixth-placed finish.
Neither manager will have been especially happy with the first half - it was almost a complete non-event in the teeming Stamford Bridge rain.
If the Chelsea players were trying to prove they were worth a place in next Saturday's showpiece Wembley FA Cup final, it was hard to tell.
Everton were easily matching their more illustrious opponents and went close when Vaughan's 25-yard dipping volley forced Cech to pluck the ball out of the air.
Salomon Kalou looked Chelsea's main threat as he began to cut in from the left and cause Tony Hibbert a few problems.
Everton goalscorer Vaughan (right) with Chelsea's Mikel Jon Obi
He drove inside and forced Tim Howard to save low down, before the Ivorian found Cole inside the area, the England man seeing his left-foot shot roll inches wide of the post.
Jose Mourinho looked appalled by his side's display - and his mood would have darkened even further just five minutes after the restart.
With Chelsea on the offensive, Osman carried the ball into the hosts' half and brilliantly delayed his pass until he could pick out Vaughan at the far post, left unmarked to slot past Cech.
Moments later it could have been game over, Cech making a world-class one-handed save to parry Vaughan's acrobatic volley from Arteta's cross.
But just as the visitors seemed to be dominating, up popped Drogba to rescue his side with a leveller.
The lively Wright-Phillips found room down the right and his clever pull-back was met perfectly by Drogba, arcing the ball away from Howard's despairing dive and into the far corner.
With parity restored Chelsea stepped up a gear, Wayne Bridge forcing Howard to save and substitute Ben Sahar drilling a shot straight at the American from 20 yards.
But Everton should have won it at the death, James McFadden's close-range strike crazily ruled offside by a linesman even though it was James Beattie in an offside position and he was nowhere near the ball.
Chelsea manager Jose Mourinho on equalling the record:
"It's nice to have levelled it, but no more than that really.
"We are in football to win trophies and championships, not to break records.
"It's an honour to have the record with a historical Liverpool team of the late 1970s and early 1980s. The culture of being strong at home, we need to keep."
Everton boss David Moyes on finishing sixth:
"This is disappointing, we thought we were going to finish fifth.
"We were incensed by the decision not to award a foul against Paulo Ferreira in the build-up to their goal, it was a terrible decision and it was wrong.
"You have to work really hard for what you get at Chelsea and Petr Cech made a great save when we were 1-0 up. But we played really well today."
Chelsea: Cech, Ferreira, Boulahrouz, Terry, Bridge (Hutchinson 89), Lampard, Mikel (Morais 74), Joe Cole (Sahar 70), Wright-Phillips, Drogba, Kalou.
Subs Not Used: Hilario, Lee Sawyer.
Booked: Bridge, Joe Cole.
Goals: Drogba 57.
Everton: Howard, Hibbert, Yobo, Stubbs, Lescott, Osman, Fernandes (McFadden 77), Neville, Carsley, Arteta, Vaughan (Beattie 72).
Subs Not Used: Turner, Naysmith, Nuno Valente.
Goals: Vaughan 50.
Ref: M Halsey (Lancashire).