Mensah scores and sent off in Ghana success
The giant defender, who has played just one league game since returning to Olympique Lyon in France this season, bundled home the ball from close range following a corner after 25 minutes.
He latched on to a flick-on from Asamoah Gyan and deflected the ball into the net off his knee as the Botswana defence froze.
But Mensah then received the first red card of the tournament, co-hosted by Equatorial Guinea and Gabon, in the 66th minute when he knocked over Botswana striker Jerome Ramatlhakwana in a chase for the ball on the counter-attack.
Ramatlhakwana had been released by a long clearance out of defence and and Mensah caught him on the edge of the penalty area and was sent off for a professional foul.
Mensah has now scored nine goals in his senior career, three of them for his country.
He missed the last Nations Cup, in which Ghana finished runners-up, because of injury and will now sit out Ghana's next Group D match against Mali in Franceville on Saturday.
Botswana qualified as one of the surprise packages of the preliminaries and enhanced their reputation with a solid defensive performance.
Veteran Ghana defender John Paintsil spectacularly hooked the ball clear off his line early in the second half after an attack by Botswana down the wing delivered a cross that Moemedi Moatlhaping headed goalwards.