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.
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.
