In what was a competitive but cagey game played in fierce heat, the Clever Boys were the more creative side and deserved the victory, secured by a first half goal from midfielder Sibusiso Vilakazi.
Wits dominated the first half and took the lead on 26 minutes, shortly after a water break. Right-back Sipho Mngomezulu played a long ball into the Cosmos penalty area and the home side's defence somehow allowed an unmarked Vilakazi to bring the ball under control, and he beat Avril Phali from close range.
Vilakazi almost set up a second goal for Wits a short while later, winning a header at the back post to play a dangerous ball across the face of goal for Sifiso Vilakazi to get onto, but this time Ezenkosi's defence did just enough.
Cosmos had a great chance to make it 1-1 10 minutes before half-time: a terrific through ball from Bamuza Sono put Siyabonga Nontshinga one-on-one with the goalkeeper, but he tried to round Energy Murambadoro, rather than simply lobbing the Wits gloveman, and missed out on the opportunity.
Ezenkosi asked a lot more questions of Wits' defence in the second half, but the visitors remained dangerous on attack, and Phali had to make a good save to keep out a Ryan Chapman effort on 68 minutes.
Phali made another excellent save 13 minutes later, diving bravely at the feet of Sibusiso Vilakazi after Wits sprung the home side's offside trap.
The missed opportunity did not cost Wits, however, as they held out for the 1-0 win and join Santos, Pirates and Golden in Monday's semifinal draw.