Senegal had 30 attempts at goal against their hosts but it took until the final seconds for Mane to seal victory with a 25-yard free-kick that curled around the wall and into the bottom corner of the net.
The 2021 winners, who recently parted ways with long-time coach Aliou Cisse, join Burkina Faso as the qualified nations from Group L in the 24-team finals with two more matches to play in the November international window.
Ghana slumped to a 2-0 loss against Sudan in neutral Libya as Ahmed Al-Tash and Mohamed Abdel Rahman scored within three minutes of each other in the second half.
Ghana look set to miss a first continental finals since 2004 having managed a paltry two points from their four games, five adrift of the qualification positions with a difficult trip to pool leaders Angola next up.
Democratic Republic of Congo qualified following a 2-0 victory in Tanzania as Meschack Elia scored a brace, his first in the 88th minute and the second deep into stoppage time. They have a full haul of 12 points from their four games.
Coach Avram Grant’s Zambia claimed an important 1-0 win over Chad to take a big step towards qualification thanks to Kennedy Musonda’s goal 20 minutes from the end.
Gabon are also on course after a 2-0 win over Lesotho in neutral Durban, with Shavy Babicka and Rody Effaghe grabbing goals in the second half.
Uganda beat South Sudan 2-1 in Juba and their qualification will be confirmed later on Tuesday if South Africa win in the Congo.
Rwanda beat Benin 2-1 in Group D with the other fixture on Tuesday between Libya and Nigeria postponed after the latter refused to play over travel chaos in the build-up.
Hosts Morocco, Cameroon and Algeria are the other teams already in the finals line-up.