Joseph Douhadji was handed his debut for the home side, while Themba Ndlovu, Mogakolodi Ngele and Robert Ng'ambi returned from their last-gasp 1-0 defeat to Orlando Pirates.
Amavarara could've gone ahead inside the fourth minute when a poor clearance landed in the path of Sihle Maso, who failed to beat Muleka from point-blank range.
The hosts immediately went up at the other end and found a golden chance from Ngele's low cross, however Xolani Ndlovu shot straight at Khanya Mini.
Leopards increased their tempo nine minutes in and hit the front after Lesedi Kapinga latched onto a through ball from the right to fire into the far corner.
Themba Ndlovu gave the home side a second goal in the 25th minute when he broke through the defensive line to find the top left-hand corner.
Lidoda Duvha looked to finish the half with more goals, but Ngele's free-kick sailed into the side-netting with Xolani Ndlovu unable to get the better of Mini from inside the area.
The home side were off to a tentative start in the second period, with Amavarara not keen to concede more thus got loads of bodies behind the ball.
Ngele, who netted the opener in the previous round, went close to giving his side a third but his low shot from outside the area went wide 12 minutes later.
Lidoda Duvha coach Alan Clark responded with their first substitution after the 60th minute, as Douhadji made way for Chris Katjiuka at the back.
Khakha Velaphi should've given the visitors a lifeline when he got on the end of a through ball in the area only to fire wide of the near post with 78 minutes gone.
Leopards had chances for more in the remaining minutes, but their first-half goals was enough to ensure quarter-final progression.
Leopards:Muleka, Mudau, Matlaba, Ndou, Douhadji, Abubakar, Ng'ambi, Ndlovu, Ndlovu, Ngele, Kapinga.
Subs: Maduwa, Ndawonde, Makgetlwa, Mokoena, Mncwango, Katjiukua, Khutlang.
Amavarara: Mini, Ntsane, Mzazi, Kelepu, Twani, Gocina, Tonisi, Mbenyana, Maso, Gxaba, Lafota.
Subs: Gwangwa, Nqumashe, Thwani, Velaphi, Sizane, Dayimani, Mambeshu.