The Lyon defender scored in the 24th minute, when he helped in a corner from the left taken by Emmanuel Agyemang Badu, to bring some life to an otherwise dull start to this Group D clash.
The Botswana defenders, who before then looked organised and fired-up, were guilty of ball watching for the experienced Mensah to guide the ball home with his knee after Asamoah Gyan appeared to help on the corner with his head.
However, Mensah saw red in the 65th minute when he committed a professional foul on striker Jerome Ramatlhakwana, who was clear on Ghana goal after a blunder by Anthony Annan. The Zebras, now with a numerical advantage, finished the stronger, but they failed to restore parity.
"We controlled the game until the red card for Mensah," said Ghana coach Goran Stevanovic. "We created some chances but they closed down the game with ten men behind the ball. All the same, it was very important that we won our first game of a tournament that has already produced some surprises."
Botswana coach Stanley Tshosane admitted his team played defensively at the start and he was satisfied with the overall performance, even though he would have wished for a better result. "I'm not happy with the result, but with the performance especially in the second half," said Tshosane. "We were cautious at the beginning because we have never played Ghana before at this level and so we preferred to study how they play."
Botswana, who were cagey and defensive in the first half, were more adventurous after the interval. This approach almost paid dividends six minutes into the second half when they came close to drawing level, but Ramatlhakwana's header from inside the box was saved acrobatically on the goal line by Rennes defender John Boye.
Ghana handed Nations Cup debuts to goalkeeper Adam Kwarasey, Boye and Jordan Ayew with usual right back Samuel Inkoom preferred for the team's problematic left-back position. For debutants Botswana, the only surprise on the starting XI was Phengo Mongala, who was picked ahead of Abendico Powell, a late replacement for the injured Joel Mongorosi.
The Zebras were also missing their most experienced player in Dipsy Selolwane, who bagged two bookings during the qualifiers and so was on a one-match suspension.
The Black Stars enjoyed more of the ball in the first quarter-hour, but hardly threatened the goal of their opponents, who packed their defence. In the opening moments of the game, the lively Andre Ayew was thrice felled around the Botswana box as Ghana went in search of an early opening goal.
After Mensah's decisive goal Ghana continued to dictate the pace of the game. In the 32nd minute, Botswana defender Ndiyapo Letsholathebe literally put his body on the line to keep out Sulley Muntari's full-blooded shot from the edge of the box.
Six minutes later, Jordan Ayew was set up by Muntari at the top of the Botswana box, but his shot was weak and so did not trouble Botswana's 35-year-old goalkeeper, Modiri Marumo. Ghana were now more daring and soon after Gyan found his way through the packed Botswana defence, but his shot flew off target.