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. 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.
Algeria host Burkina Faso on November 19 in the return match and the overall winners qualify for Brazil 2014. The home team suffered an early blow when defender Bakary Kone was forced to retire with a thigh injury and Steeve Yago came on.
But as the opening half passed the half hour, 2013 CAF 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 Essaid Belkalem from English second-tier club Watford floored him with a wild tackle.
But Adi Mbolhi dived left to make a two-hand save from a hard Bance kick. However, the Burkinabe did take the lead two minutes into stoppage time as Pitroipa met a Jean-Noel Lingani cross and his header landed in the far corner.
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 play of the early stages 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 once more 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.