The Chelsea striker opened the scoring after 16 minutes and a late own goal from Bakary Kone sealed the victory as the tournament favourites, having beaten Sudan in their opening match, guaranteed themselves a top-two finish in Group B.
This loss signals the end of Burkina Faso's hopes, as they remain rooted to the foot of the standings having also lost 2-1 to Angola.
Burkina Faso had their chances in a entertaining encounter in Malabo, with Beli Moumouni Dagano hitting the crossbar late on and Alain Traore a regular threat with his long-range thunderbolts, but Ivory Coast also had a raft of opportunities and they will hope their finishing improves in the latter rounds as they seek only their second Nations Cup title.
Burkina Faso felt they should have been awarded a sixth-minute penalty when Jonathan Pitroipa went down in the area following a challenge from Souleymane Bamba, but that was waved away and 10 minutes later they found themselves trailing.
Didier Drogba sent over a cross from near the right corner flag which, after Yaya Toure had just failed to collect, found its way through to Kalou in the middle and he drilled home a sweetly-struck left-footed strike past Daouda Diakite.
Ivory Coast would have hoped that was the start of a comfortable night's work, but Burkina Faso proved no easy opponents and Traore twice went close with efforts from the edge of the box as they sought an equaliser.
Ivory Coast almost doubled their lead on the stroke of half-time when Diakite did well to deny Drogba and Gervinho in quick succession, although the Arsenal man's effort would not have counted anyway due to offside.
Gervinho spurned a decent opening to increase his side's advantage at the start of the second half, before the left-footed Traore saw two good efforts saved by Boubacar Barry.
Gervinho blazed an even better opportunity over when unmarked six yards out in the 60th minute and soon after he produced a good parry out of Diakite after bending a shot around a defender.
Kalou helped set Gervinho up for that chance but the Chelsea man suffered what appeared to be a hamstring injury in the build-up and he was replaced by Max Gradel soon after.
Ivory Coast survived a huge let-off in the 72nd minute when Dagano got on the end of a superb deep cross from Charles Kabore before powering a header goalward, but fortunately for the Elephants the ball smashed against the crossbar and went over.
Dagano then saw another header comfortably saved by Barry before Ivory Coast sealed the win with an own goal in the 82nd minute.
Arthur Boka launched a long ball forward into the Burkina Faso area and, as team-mates Kone and Mamadou Tall challenged for the ball without an Ivory Coast player involved, the former succeeded only in sending a looping header over Diakite and into his own net.
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