Algeria won the eagerly-anticipated north African derby against Morocco but needed a penalty to take the points.
The Ivorians are now effectively one win away from booking a place at next January's tournament, to be co-hosted by Equatorial Guinea and Gabon, after having to come from behind to beat Benin 2-1 in Accra.
They have won all three matches to open up a five point lead in Group H.
Forced to play the match in neighbouring Ghana because of the security situation in their own country, the Ivorians were behind when a slip by goalkeeper Daniel Yeboah allowed a long range shot from Seidath Tchomogo to trickle in.
But Drogba, who has spoken before the match of the importance of a win to lift the mood of the country, equalised on the stroke of half-time and headed home the winner 15 minutes from the end.
Ghana, who face a dash from central Africa to London for their Tuesday friendly against England at Wembley, kept ahead in their group with a comfortable 3-0 away win in Brazzaville against Congo.
Prince Tagoe opened the scoring after 20 minutes and Dominic Adiyiah, who also plays at Partizan Belgrade, nodded in a John Paintsil cross before the break.
Sulley Muntari, who had been doubtful before the match, scored within three minutes of coming on as second half substitute.
Sudan, who beat Swaziland 3-0 at home on Sunday, are just behind Ghana in Group I on goal difference.
NIGERIA ROMP
Peter Utaka and Ike Uche got two goals each for Nigeria in their 4-0 romp over Ethiopia in Abuja but they are still one point behind Guinea in Group B.
Guinea equalised only five minutes from time through Mamadou Bah to draw 1-1 in Madagascar on Sunday.
Goals from James Chamanga and Emmanuel Mayuka ensured a 2-0 win for Zambia away in Mozambique in Group C. They take over top place from Libya, who play on Monday against the Comoros Islands in neutral Mali.
Hassan Yebda's fifth minute penalty gave Algeria a 1-0 win in the derby against Morocco, played at Annaba. It was a major boost for the 2010 World Cup finalists, who were bottom of the group before Sunday's game.
