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.
