The Super Eagles almost squandered a three-goal lead as Montassar Talbi pulled a goal back in the 74th minute and Ali Abdi converted a penalty in the 87th.
Osimhen missed two good chances with headers early on as Nigeria made a good start, but he finally made one count before the break with a header from Ademola Lookman’s cross.
It was just Osimhen’s second Africa Cup goal in his career.
Nigeria were cruising after the break when Lookman set up Wilfred Ndidi for the second goal, then Osimhen set up Lookman in the 67th.
The Carthage Eagles somehow found a way back and threatened an unlikely comeback after Bright Osayi-Samuel was penalised for handball after a VAR check.
Talbi scored the penalty and fans in the Complexe Sportif de Fès set off flares in delight.
Tunisia captain Ferjani Sassi headed just wide in stoppage time and the Nigeria players ultimately showed enough experience to see out the win.
Nigeria are assured of their place in the last 16 with a maximum six points in Group C, three ahead of Tunisia while Tanzania and Uganda have a point each.
The top two progress automatically to the last 16, along with the best third-place finishers.
Uganda and Tanzania – two of the three co-hosts of the next tournament with Kenya – drew 1-1 in a frenetic early Group C game, a result that did little for either after opening losses.
Uganda’s Allan Okello missed the chance to win the game when he struck a penalty well over in stoppage time, after a rain deluge had drenched the players and sent fans scampering for cover.
Tanzania’s Simon Msuva put the Taifa Stars ahead with a penalty early in the second half, then Uche Ikpeazu equalised with a diving header late, just three minutes after coming on as a substitute.
That goal prompted the heavens to open, adding to the chaos of the final minutes as both sides pushed for a win to boost their hopes of progress.
The 17-year-old French-born Ibrahim Mbaye rejuvenated his team and provided the impetus for Sadio Mané to score in a 1-1 draw with Congo.
Senegal dominated possession and chances with Nicolas Jackson and Mané again missing good opportunities, as they did in the opening win against Botswana.
Cédric Bakambu pounced on the rebound to open the scoring for Congo in the 61st minute after Édouard Mendy had saved Théo Bongonda’s initial shot.
Senegal coach Pape Thiaw reacted by sending on Mbaye and the 17-year-old made an immediate impact, though not before Bongonda could have made it 2-0 on a counter attack.
Mbaye ran at the Congolese defenders and had an effort saved by goalkeeper Lionel Mpasi, with Mané tucking away the rebound to level in the 69th.
“We are not completely unhappy as the Afcon is always tough,” Mané said. “It was a tough match and in this competition you cannot underrate any team.”
Mbaye was left free when Congo defender Arthur Masuaku pulled up with what looked like an Achilles injury.
Masuaku was unable to continue, but Mbaye, who plays for Paris Saint-Germain, could get more opportunities to shine against Benin in Senegal’s final group game.
Earlier, Yohan Roche’s first-half strike was enough for Benin to beat Botswana 1-0 for their first ever Africa Cup win.
“It is a source of immense pride,” Roche said. “We were aware of the people’s expectations, and we managed to stay focused.”
After two games, Senegal lead Group D with four points, ahead of Congo on goal difference, with Benin third on three, followed by Botswana with none. (AP)
