Bernard Parker gave the Rabbits a first half lead before Fanteni struck twice to the Urban Warriors a crucial away win.
The hosts started well and deservedly went ahead in the 26th minute when Parker headed home from Joseph Kibe.
The Rabbits continued to play the better football until the end of the half.
Ajax came back stronger in the second half and were rewarded in the 75th minute when substitute Fanteni slotted home from a pass by Mkhanyiseli Siwahla.
The visitors secured the points when they were awarded a penalty 10 minutes from the end when Siwahla was fouled inside the area and Fanteni stepped up to coolly slot home.
The Rabbits take on Platinum Stars on September 16 in Mafikeng, while Ajax Cape Town hosts Jomo Cosmos at Athlone Stadium on the same day.