Christopher Netshidzivhe scored the goal.
The result marked Leopards' first win in the league after 14 games and gives them an outside chance of avoiding relegation.
The first half was a dour affair which did not dish up a single shot on target by either side.
The hosts suffered a major set back in as early as the fourth minute when they were reduced to 10 men after Lebo Kekana was given his marching orders for a challenge on Ashraf Hendricks.
The decision looked rather harsh as the player's momentum had carried him through and there did not appear to be any intent.
However, despite their numerical disadvantage, Lidoda Duhva looked the better team in the first half against a woefully poor Wits side.
The first real clear goal-scoring chance of the game came in the 37th minute when Mulondo Sikhwivhilu burst down the right wing, but in trying to curl the ball with the outside of his boot past Moeneeb Josephs, the effort flew inches wide of the far post.
The Clever Boys' came close to scoring just before the break when Dale Studzinski's header picked out Richard Rantjie, whose angled shot was just off target.
Leopards again looked the stronger team in the second half as Wits put in a lacklustre display.
A good example of their intensiveness came in the 63rd minute when Studzinski tried to pick out a team-mate rather than shooting at goal himself.
Three minutes later The Students paid the price for their poor performance when Christopher Netshidzivhe struck with a superb 25-yard free kick in the 66th minute.
Eleven minutes later Wits were also reduced to 10 men when Studzinski received his marching orders.
The remainder of the game saw Leopards remain on the offensive and while they had several chances to add to the score-line, the solitary goal proved enough to grab all three points.