Lebohang Mokoena opened the scoring for Sundowns in the 30th minute, and the game went into half-time with the home side 1-0 up.
But with Maartin Stekelenburg's advice to his team to "have fun" and play the ball, Ajax came back through Terror Fanteni and Nhlanhla Vilakazi.
Punch Masenamela got his first goal for Sundowns on 83 minutes to level the scores, but Khama Billiat got the winner in injury time.
The Brazilians would have returned to the top of the log had they claimed all three points. Instead they are placed third on 10 points, two adrift of leaders Kaizer Chiefs. Ajax are fourth, also on 10 points.
Sundowns dominated the first half and opened the scoring on 30 minutes when Mokoena curled home a left-footed shot through a crowd of defenders. The Sundowns attacker capitalised on a rebound after Sean Roberts had saved from a snap shot by Katlego Mphela.
Franklin Cale almost doubled the hosts’ lead six minutes later with a dipping volley that crashed back into play off the crossbar.
The second half was much more evenly matched and Ajax started their comeback on 62 minutes when Fanteni controlled a long ball out of defence, pushed into space and unleashed a fierce shot that Wayne Sandilands was helpless to stop.
Just over 10 minutes later the Cape side went in front for the first time. Lebogang Manyama unleashed a long range shot that was saved by Sandilands – however, the ball rolled free in the penalty area and Vilakazi had an easy task to fire the ball home and make it 2-1.
With eight minutes to play, Sundowns set up the game for a dramatic finale by making it 2-2. Masenamela picked up the ball on the edge of the Ajax penalty area and fired a left-footed drive into the back of the net. Roberts got a hand on the shot, but it had too much power to be pushed aside.
The game seemed to be there for Sundowns to win in the dying minutes, but instead it was the Urban Warriors who provided the late heroics.
In the first minute of injury time Ajax sliced open the Brazilians’ defence and Billiat – who had been completely anonymous throughout the regulation 90 – popped up to slide home the winning goal at 3-2.