Ezenkosi scored three second half goals to eventually beat SAB side United Rovers 3-0 and book their place in the Last 16 of this year's Nedbank Cup.
Wiseman Maluleke netted a brace, while Matsiele Sono sealed Cosmos' progression with his side's third goal at the Tsakane Stadium in Brakpan.
Ezenkosi were joined in the Last 16 by Bloemfontein Celtic. Deon Hotto and substitute Vuyisile Wana netted as Phunya Sele Sele registered a 2-0 win over ABC Motsepe League side Buya Msuthu at the Moruleng Stadium.
PSL strugglers Baroka FC registered a comfortable 2-0 win away to FC Cape Town on the same day.
Ndivhuwo Ravhuali opened the scoring with a well-taken penalty, before Richard Matloga put the game to bed with a towering header in the closing stages of the game.
In Durban, ABC Motsepe League side KwaDukuza United pulled of a major upset by knocking Ajax Cape Town out of the tournament on penalties.
The two sides could not be separated after regulation, and after 120 minutes of football, KwaDukuza and Ajax were locked at 2-2, forcing the encounter to be decided on penalties.
KwaDukuza won the match 4-3, and therefore became the first club to beat a PSL outfit.