A last-gasp penalty from Oswin Appollis proved decisive in Marrakech, ensuring the 1996 champions advanced as second-placed finishers in the group with six points.
They ended just one point adrift of group leaders Egypt, who were already assured of progression and played out a goalless draw with Angola in the other Group B encounter.
Bafana Bafana twice moved in front during an action-packed contest, with Tshepang Moremi opening the scoring before Lyle Foster later restored their advantage. However, Zimbabwe responded on both occasions, first through Tawanda Maswanshise and then via an unfortunate own goal by Aubrey Modiba.
With the game seemingly destined to finish level, late drama unfolded when a VAR check spotted a handball on the goal line by Zimbabwe midfielder Marvelous Nakamba.
Appollis calmly dispatched the resulting spot kick, netting his second goal of the competition and securing victory for South Africa.
The result sends Hugo Broos’ side into the knockout phase, where they will meet the second-placed team from Group F in the Round of 16.
Meanwhile, Egypt finished as Group B winners at the claiming seven points after a 0-0 draw with Angola on Monday.
