Mali finished second in Group D, behind Ghana, and set up a meeting on Sunday with co-hosts Gabon in Libreville.
Mali proved unconvincing in the match, watched by a 5,000 crowd, but Guinea's failure to beat Ghana in the other group game in Franceville ensured their progress.
He played a pass out wide and delayed his run into Mali's penalty box to arrive at the right moment and sweep the return ball home.
But Mali's pressure paid off six minutes later with Abdou Traore turning the defence before providing a cross that Modibo Maiga forcefully headed goal ward. Botswana goalkeeper Modiri Marumo could only parry it right to Garra Dembele who hammered home from close range.
"It was a huge lift for us to score then and the win means everything. We have a chance now to go all the way," Dembele told reporters.
Keita's goal was set up by his persistence on the edge of the penalty area as he probed an opening and then a return pass from Cheick Tidiane Diabate to perfectly place the ball into the roof of the net.
They then had to wait for a full five minutes until Guinea's failure to beat Ghana was confirmed.
"I was full of anguish at the end of the game having to wait. There is always this uncertainty, so I preferred to go back to the change room and wait patiently for the a outcome," said Mali coach Alain Giresse, who was in a similar situation at the last Nations Cup two years when in charge of Gabon.
"We had a chance to compete in a tournament we never even dreamt of going to. That will be our history for the rest of our lives."
