There was also a setback for Morocco, whose coach Eric Gerets' job is now under severe threat after a 1-1 draw away at lowly-ranked Gambia.
Drogba and his Chelsea team-mate Salomon Kalou scored for the Ivory Coast as they beat Tanzania 2-0 at home less than a week after changing their coach.
It was the first game for the pair since the Champions League final against Bayern Munich on May 19 and a winning debut for new Ivorian coach Sabir Lamouchi, whose surprise appointment on Monday had been widely criticised.
Algeria, Cameroon, Senegal and Tunisia also triumphed on Saturday as 11 matches were played at the start of the group phase. It followed two matches on Friday and there will be seven more on Sunday.
Both Senegal and Tunisia came from behind to win 3-1 at home in their respective group games.
Senegal overcame Liberia in Dakar while Tunisia beat Equatorial Guinea in Monastir, where a few thousand were allowed to watch the game after months of barring spectators from football in the north African country because of security concerns.
Cameroon, who have been to an African record six World Cups, needed a second half penalty from Eric Choupo Moting in an unconvincing win over the Democratic Republic of Congo in Yaounde.
Striker El Arabi Soudani scored twice as Algeria had a routine 4-0 triumph over Rwanda in Blida.
Zambia lost their first competitive game since February's Africa Cup of Nations success as Sudan beat them 2-0 in Khartoum.
Second half goals from Mohamed Tahir and Seif Masawi brought the Zambians back to earth with a sudden jolt and makes next Saturday's home match against Ghana vital for the continent's champions.
The Central African Republic recorded a first ever World Cup qualifying win with a 2-0 home triumph over Botswana in Bangui but Burkina Faso and Kenya dropped points with home draws.
There will be another round of qualifiers next weekend with all 40 countries in the group phase involved.
