The Pharaohs had lost their opening two matches, but came good in Gaborone thanks to two second half goals from Mohamed El-Neny and Mohamed Salah.
Despite the win, Egypt remain in third place in Group G, three points behind Senegal and Tunisia and having played a game more. Botswana are bottom with three losses from their three games.
In another qualifier on Friday night, Angola were held to a goalless draw by Lesotho in Maseru.
That leaves Palancas Negras bottom of Group C with one point, one less than third-placed Lesotho.