Drogba converted the fifth-minute spot kick, which was followed by an own goal from Lamine Sane nine minutes later and then a breakaway effort from Salomon Kalou just after the break in Abidjan.
Senegal's Papiss Cisse pulled one back five minutes into stoppage time with the last kick of the game.
Although Drogba continued his talismanic influence on the Ivorians, who qualified for the previous two World Cups in Germany in 2006 and South Africa in 2010, Italian-based midfielder Gervinho was central to all the home goals.
He was brought down by Cheikh Kouyate to earn the penalty and then Sane contrived to turn Gervinho's probing ball into his own net.
Five minutes after halftime, Gervinho slipped a perfectly-weighted pass between two defenders to allow Kalou to run away and tuck the ball home.
But the Ivorians collapsed to the turf at the final whistle after sloppy defending allowed Cisse to sneak in at the back post and convert a cross as the referee was about to blow the final whistle, in a rare highlight for the visitors .
The return match is being played in Casablanca, Morocco on Nov. 16 because Senegal have a ban on their home ground in Dakar.