Chelsea's mid-season slump continued after a late equaliser from substitute Jermaine Beckford earned Everton a 1-1 draw at Stamford Bridge.
The result left Chelsea with just one win in their last six league games and is sure to pile more pressure on under-fire boss Carlo Ancelotti.
Everton almost shocked Chelsea in the opening few seconds when Louis Saha's shot from 20-yards forced Petr Cech into a fine save at his near post.
Moments later Leighton Baines fired high over the bar with Chelsea still struggling to get to grips with the high-octane opening from the visitors.
Chelsea's first real attack arrived in the 12th minute when Drogba put Nicolas Anelka ahead of the visiting defence.
The former France international's shot was blocked by Phil Jagielka to the relief of Everton keeper Tim Howard.
Chelsea felt they should have had a penalty when Jose Bosingwa's cross appeared to hit the hand of Sylvain Distin but referee Lee Probert was not impressed.
Anelka tried his luck from 20 yards in the 20th minute but his shot flashed wide of Howard's right-hand upright.
The woodwork prevented Chelsea from taking the lead in the 27th minute when captain John Terry, back after a sciatic nerve injury, sent a shot against the bar.
It followed a sustained spell of pressure from the home side which resulted in a spat between Florent Malouda and Everton's Phil Neville.
Neville twice lunged at the France international from behind causing the Chelsea player to retaliate. But to Neville's credit he got up immediately and referee Probert took no action.
However Neville was booked minutes later when he fouled Drogba on the halfway line, much to the consternation of Everton manager David Moyes.
Chelsea wasted another chance in the 37th minute when Salomon Kalou headed wide from a corner by Malouda.
Chelsea eventually broke the deadlock from the penalty spot in the 41st minute when Anelka pounced on a poor back pass from Neville and was brought down by Howard.
The Everton keeper was spared a red card by Probert, who opted for a yellow instead, but Howard could do nothing to stop Drogba's fierce penalty from giving the Blues the lead.
Everton began the second half brightly but Jack Rodwell's accuracy from 20 yards left a lot to be desired.
His 51st minute effort was well wide of the target but Everton were showing signs of a revival.
Chelsea were still a danger on the counter-attack and Drogba should have done better when Bosingwa's cross allowed him time to turn inside the penalty area.
The Ivory Coast striker's shot was easily saved by Howard and Everton almost levelled in the 61st minute when a cross from Leighton Baines was headed against the inside of the post by Rodwell.
Chelsea were now having to defend in numbers with Everton controlling the game.
Two minutes later a corner from Baines was headed on by Jagielka but the ball dropped agonisingly wide of the far post.
There was controversy in the 72nd minute when Terry and Tim Cahill squared up to each other after the Everton midfielder had collided with Cech.
Players from both sides got involved in the aftermath and Terry accused Cahill of deliberately leading with his studs.
Cech, who has to wear a protective helmet following a fractured skull suffered four years ago, was given lengthy treatment to his forehead.
Everton grabbed a deserved leveller four minutes from time when a cross from Baines was headed back across the six-yard box for substitute Beckford to head home.
