Relebohile Mofokeng looked to have won the game for the home side with his late striker, but Ogam capitalised on a defensive error from Olwethu Makhanya to stun Bafana Bafana and deal a blow to their qualification hopes.
Bafana Bafana took an early lead through an own goal by Kenya defender Amos Wanjala, but the visitors equalised through returning captain Michael Olunga, whose penalty just before halftime took him to 35 international goals and level with William Ouma among Kenya’s all-time leading scorers.
Bafana still top the pool with seven points from their first three matches, one point ahead of Guinea, where they travel on the final round of fixtures in March.
Kenya have already secured their place at the continental finals as co-hosts, meaning only one other team from the group will advance to the tournament, which runs from 19 June to 17 July next year.
Some good build-up play from South Africa resulted in a deep cross into the box, but Iqraam Rayners’ header was straight at Kenya goalkeeper Byrne Omondi.
Tshepang Moremi missed an even better chance shortly afterwards when Mofokeng teed up the winger, only for him to scoop the ball over the crossbar.
Bafana went ahead in the ninth minute as the frantic opening to the match continued, and it was a horror moment for Wanjala.
Kenya tried to play out from the back, but as Mofokeng put pressure on Wanjala, the defender attempted to pass back to Omondi. His effort was too strong, however, and ended up in the back of the net.
It was the 13th successive Africa Cup of Nations qualifier in which Bafana had found the back of the net.
Kenya were handed the chance to level the score when a deep cross was headed back towards the six-yard box and the ball struck the outstretched hand of Bafana right-back Thabang Matuludi. It was a clear penalty.
Olunga made no mistake, confidently dispatching the spot-kick into the bottom-left corner of the net.
Rayners should have restored the lead on the stroke of halftime, but his shot from point-blank range was saved by Omondi.
The striker had another excellent opportunity early in the second half, but again could not beat Omondi when one-on-one with the goalkeeper.
Oswin Appollis teed up Mofokeng for a shooting chance, but his placed effort came back off the post as the hosts continued to be frustrated.
Lyle Foster headed over the bar with a free header, but the goal Bafana desperately needed finally arrived in the 77th minute.
Mofokeng floated a ball towards Foster at the back post and then raced into the box to receive the return pass, volleying into the top corner of the net.
The home side would have believed that was the winner, until hesitant defending from Makhanya allowed Ogan to run at goal and he supplied a superb finish into the bottom corner of the net.