Full back Thabo Matlaba drove in a long-range shot in the 33rd minute before a second-half goal by Bernard Parker pushed the home team to the top of the group with five points from three matches.
Ethiopia, who have four points, can go back to first place if they beat bottom team Botswana in Addis Ababa on Sunday. Central African Republic are third with three points.
South Africa dominated early on against their physically-imposing opponents but Swedish-based striker Tokelo Rantie missed several chances.
Central African Republic's French-based goalkeeper Geoffrey Lembet then made two excellent saves in the second period.
The visitors had an effort disallowed after halftime when substitute Vivien Mabide punched the ball in following a corner. The referee awarded a goal only to change his mind after consulting his linesman.
The two countries meet again in Bangui in June.
- Sports & Recreation
- Central African Republic
- Bernard Parker