The Chelsea striker also missed penalty in the game, while Manchester City midfielder Yaya Toure was on target to set up a last four meeting with either Mali or the other co-hosts Gabon.
Ivory Coast were back to full strength after resting a host of stars for their final group game, and that added up to danger for Equatorial Guinea, who had Laurence Doe back from suspension.
After a slow start, Ivory Coast began to step up the pace of their play midway through the first half, and Gervinho, Toure and Max Gradelthreatened to open the scoring before Didier Zokora was brought down to bring about a penalty.
Goalkeeper Danilo kept out Drogba's spot-kick, but could not prevent him firing the Ivorians ahead in the 36th minute after a defensive mix-up.
A thrilling run from Gervinho midway through the second half saw the Arsenal forward wind his way into a threatening position but he lost control at the critical moment, when he should have been firing the ball past Danilo.
It did not matter because Ivory Coast doubled their lead after 70 minutes when Drogba headed in a fine second goal from Toure's free-kick.
With the former Barcelona man pulling the strings in midfield and Drogba a serious menace to any defence in this competition, it is little wonder many have tipped Ivory Coast from before the tournament.
Toure showed his pedigree again from a set-piece with eight minutes remaining when he curled home a fine strike to make absolutely certain of victory.