The 35-year-old striker converted a penalty after five minutes at Stade Felix Houphouet-Boigny, Lamine Sane conceded an own goal and Salomon Kalou netted soon after half-time.
But Papiss Cisse pulled one goal back six minutes into stoppage time after the ball slipped from the grasp of goalkeeper Boubacar Barry.
Aristide Bance converted a disputed late penalty to give Burkina Faso a 3-2 win over Algeria in Ouagadougou in another Africa zone play-off first leg.
Crowd violence when Senegal hosted Ivory Coast last year in an Africa Cup of Nations qualifier led to a one-year ban on matches in Dakar.
So on top of trying to wipe out a two-goal deficit, the Senegalese must play the November 16 second leg in Moroccan city Casablanca.
Turkey-based Drogba put the Ivorians ahead after Cheikhou Kouyate fouled Gervinho, whose cross nine minutes later was turned into his own net by Sane.
Roma wide man Gervinho was also involved in the third goal, sending Kalou through the middle to beat Bouna Coundoul at his near post.
Bance snatched the winner after Essaid Belkalem appeared to have his arms behind his body when the ball struck him four minutes from time.
But a Zambian assistant referee insisted it was a hand-ball and Bance made up for having an earlier spot-kick saved.
The home team suffered a blow when defender Bakary Kone was forced to come off with a thigh injury on seven minutes and Steeve Yago came off the bench.
Algeria settled quicker and mounted numerous attacks without seriously threatening to score at a near-full 40,000-seat Stade du 4 Aout.
Burkina Faso seemed nervous with one corner so over-hit by captain Charles Kabore that it went out of play on the far touchline.
But as the opening half passed the half hour, 2013 Africa Cup of Nations star Jonathan Pitroipa began to stamp his authority on a game needing a lift.
He won a penalty as the first half drew to a close when Belkalem from English second-tier club Watford floored him with a wild tackle.
But Adi Mbolhi, who appeared to move off his line too soon, dived left to make a two-hand save from a hard Bance kick.
Algeria began the second half strongly and were level on 50 minutes when Sofiane Feghouli fired past Daouda Diakite from close range after Islam Slimani set him up.
The cagey early play had given way to more adventurous football from both teams and Burkina Faso and Algeria scored again within five minutes.
Djakaridja Kone put the ‘Stallions’ ahead by powering forward from midfield and firing wide of Mbolhi into the far corner.
But a flying Carl Medjani header off a corner gave Diakite no chance and the teams remained level until Bance atoned for his earlier penalty miss.
Ethiopia play Nigeria and Tunisia meet Cameroon Sunday and Ghana face Egypt Tuesday and the five overall winners qualify for the 2014 tournament in Brazil.