Egypt - who won the tournament three times in a row from 2006 to 2010 - were in danger of not even making the competition proper after defeats to Senegal and Tunisia.
But at the third attempt they won their first Group G points, a result that brings them within four of Senegal and Tunisia.
Egypt started brightly despite the untidy pitch and Amr Gamal and Mohamed Salah both had early chances.
Salah fired a warning shot when he sharply cut inside and fired at Botswana goalkeeper Kabelo Dembe but the Egyptians had to wait until after half-time to take the lead.
Mohamed El Nenny gave them that advantage, firing into the right corner from 25 yards in outstanding fashion.
And El Nenny's strike - scored in the 56th minute - was quickly followed by a special effort from Salah.
Salah played a neat one-two with Gamal before curling to the right of the diving Dembe to seal Egypt's win.
The other Group G match saw Senegal and Tunisia play out a goalless stalemate.
Also on Friday, Lesotho and Angola could not be separated in a 0-0 draw.
