They both found the net in a 2-0 victory over Tanzania, six days after Africa’s top-ranked team made a shock coaching change.
Kalou opened the scoring after 18 minutes for the home side who drew criticism across the country after firing Francois Zahoui on Monday and replacing him with inexperienced former French international Sabri Lamouchi.
Drogba, playing his first game since scoring a match-winning penalty for Chelsea in their Champions League final win over Bayern Munich on May 19, added the second goal 15 minutes from the end.
The Ivorians, who have played at the last two World Cups, meet Morocco away in another African qualifier next weekend.
Morocco had to fight from a goal down to draw 1-1 at Gambia on Saturday.
A penalty from striker Eric Choupo Moting gave Cameroon a winning start to their World Cup qualifying campaign with a 1-0 victory over the Democratic Republic of Congo in their Group I game.
The goal came 12 minutes into the second half after the Bundesliga-based striker had been brought down but it proved to be an unimpressive win for the Indomitable Lions, whose six World Cup finals appearances is the most by an African country.
“We’ve won the game, this is a good start for us,” said coach Denis Lavagne. “But much still remains to be done, we need to press from the beginning to the end.”
Although Cameroon had most of the possession in the first half it was the visitors who created the best chances, with goalkeeper Idriss Carlos Kameni making two great saves in the 15th and 19th minutes of the game.
Cameroon next take on Libya in Sfax, Tunisia on June 10 while DR Congo host Togo in Kinshasa.
Senegal came from behind to win their opening World Cup qualifier 3-1 against Liberia as their new-look team began the process of expunging the disappointment of failure earlier this year.
The home side, who were one of the favourites at the African Nations Cup in January but collapsed in spectacular fashion, fell behind in the second minute after a goal by Francis Doe.
New Senegal captain Papiss Cisse missed a penalty before Ibrahima Balde equalised before halftime.
Substitute Dame Ndoye and French-based midfielder Sadio Mane, in only his second appearance for the team, added two more goals after the break.
Ndoye’s goal in the 71st minute came after Liberia had defender Solomon Grimes sent off.
Mane, who is in the Olympic squad for the London Games next month, made sure of the win in the 83rd minute.
Senegal, who appointed Frenchman Pierre Lechantre as coach in April only for him to walk away from the job just a few days later, play a testing away match against Uganda in African Group J qualifying next Saturday.