The rather boring match was set to end in a stalemate when Senegal were awarded a penalty in stoppage time.
Mane stepped up and powerfully slotted the ball home to secure three points for Group B favourites.
Senegal almost took the lead within a minute, as Bouna Sarr’s long-range effort was just wide of the near post.
Sadio Mane had the best chance of the first half, making a 30-yard dribble on the far-side touchline, then rounding a Zimbabwe defender, but his shot was tame and it was straight at Petros Mhari.
In the second half, Abdou Diallo came close to scoring for his side, as the centre-back headed over Sarr’s free-kick.
A point was snatched from Zimbabwe right at the very end, when Mane calmly dispatched a dramatic late penalty after Kelvin Madzongwe handled the ball in the box.