The hosts left it late in Addis Ababa, with substitute Getaneh Kebede scoring in the 88th minutes.
Thie victory leaves Ethiopia unbeaten in Group A, with seven points from three matches.
Bafana are second on five points following their 2-0 win over Central Africa Republic yesterday.
CAR are third on three points, with Botswana bottom of the group with one point.
7/6 Botswana vs Ethiopia
7/6 Central African Republic vs South Africa
14/6 Botswana vs Central African Republic
14/6 Ethiopia vs South Africa
6/9 South Africa vs Botswana
6/9 Central African Republic vs Ethiopia