Riyad Mahrez gave Algeria a 12th-minute lead at Stade de Franceville, before Zimbabwe raced into the lead through goals from Kudakwashe Mahachi and Nyasha Mushekwi.
Algeria's blushes were spared when Mahrez struck again eight minutes from full-time to make it 2-2 but Leekens was less than impressed by his side's first half performance.
"I am disappointed with the result of this first game against Zimbabwe," Leekens told beIN Sports . "The first game is always difficult.
"We scored our second goal and we could have scored more, if it were not for the brilliant performance of the Zimbabwe goalkeeper.
"We failed to press Zimbabwe when they had the ball. We came back in the second half and deserved to score more than one goal with all the opportunities we created.
"I am satisfied with the reaction of our players in the second half. We must now concentrate on Tunisia, who must be taken very seriously. We must not make the same mistakes on the road ahead."
Zimbabwe coach Callisto Pasuwa took his side into the game as underdogs but emerged feeling disappointed to have seen victory slip through his fingers.
He said: "I would say this is two points dropped."