Tashreeq Morris broke the deadlock with 11 minutes remaining in the second half when he got up to head home Mosa Lebusa's pinpoint left-wing cross.
Granwald Scott put the cherry on top with a peach of a freekick in the final minute of regulation time, sending a sweetly struck shot into the back of the net off the underside of Washington Arubi's crossbar.
It was an evenly contested game with the home side enjoying most of the possession, with Keagan Dolly and Erwin Isaacs missing the best chances for Roger de Sa's side.
But the Urban Warriors' perseverence finally paid off in the closing stages as defensive mistakes undid all of Tuks' hard work throughout the game.
The result puts Ajax in sixth place on the Absa Premiership table, level on points with fifth-placed Bloemfontein Celtic, while Tuks are not yet mathematically safe as they hover eight points above the relegation places with four matches left.