Mahrez, named African Footballer of the Year earlier this month, scored twice for Algeria but his team had to settle for a 2-2 draw with rank outsiders Zimbabwe in the opening match of a Group B double bill in Franceville.
Mane, voted the continent's third-best player, converted an early penalty to set Senegal on their way to a 2-0 win in the later game, although the continent's top ranked nation were far from convincing.
Senegal's win was as much down to the wastefulness of their opponents as anything else as Tunisia squandered chance after chance in a dominant second half performance.
"We were strong in the first half but let it slip a little after the break," said Senegal coach Aliou Cisse, whose team were among the pre-tournament favourites.
Mane's 10th minute spot kick was followed by a powerful header from Kara Mbodji that doubled the score on the half hour mark, leaving Tunisia as the first team to suffer defeat at the tournament after the first three games were drawn.
Zimbabwe were just eight minutes away from a famous win on their return to the finals after an 11-year absence before Mahrez snatched an equaliser.
On Saturday, captain Pierre-Emerick Aubameyang steered hosts Gabon ahead in the tournament's opener before they conceded a late goal to settle for a 1-1 draw against debutants Guinea-Bissau.
In the second match, Burkina Faso came back from a goal down against Cameroon as their match also ended 1-1.
The next sequence of Group B games are at the Stade Franceville on Thursday.
On Tuesday, holders Ivory Coast open the defence of their title in Group C at a new stadium in Oyem against Togo. The second game pits Democratic Republic of Congo against Morocco.
-BBC