Both goals came minutes after Drogba was introduced by Lamouchi, and the French coach has praised the game-changing qualities of his striker.
"We didn't start the game very well by conceding through a lapse in concentration. After that, we got into a lot of good positions," he told reporters at full time.
"For us it was really important to start well this tournament. To have a player like Didier Drogba in your squad, as a starter or a sub, changes a game.
"The players are very happy with the victory, but we are not yet qualified."
Ivory Coast sit second in Group C behind Colombia, who enjoyed a comfortable 3-0 win over Greece on Saturday.