The Toffees took a fortunate lead when the ball bounced into goal off the back of keeper Petr Cech's head.
But on the stroke of half-time substitute Yakubu Aiyegbeni levelled after a defensive howler from Ricardo Carvalho.
Drogba again put the hosts in front after more poor defending but then the Ivorians defensive clearance hit Louis Saha and beat Cech for another equaliser.
The North West derby at the Reebok Stadium also ended 3-3 between Bolton and Manchester City.
Ivan Klasnic gave the Wanderers the lead only for Carlos Tevez to level.
Centre half Gary Cahill restored Wanderers lead but City equalised before the break through Micah Richards.
Klasnic again gave the hosts the lead before City's Craig Bellamy saw red for a second yellow card.
If Bolton thought this signalled a valuable three points they were mistaken as Tevez earned draw specialists City a share of the spoils.
It was all square at Turf Moor were Fulham took a second half lead through Bobby Zamora only for Wade Elliot to keep the Clarets impressive home record intact.
Birmingham are in fine form at present and this continued with a narrow 1-0 home win over struggling West Ham.
Former Hammer Lee Bowyer scored the only goal against his old colleagues for his sixth strike of the campaign.
Hull had the better chances but failed to get past a defensive minded visitors.
Stoke City's entertaining 2-2 draw with Wigan was illuminated by a goal of the season contender from Figueroa and a last gasp penalty stop by Thomas Sorensen.