Camaldine Abraw gave Chiefs an early lead with a superb left-footed strike from 25 yards out after neat one-touch build-up play between Siphiwe Tshabalala and Edward Manqele.
Erick Mathoho made it 2-0 in the 24th minute with a towering header from Tshabalala's inswinging corner-kick, which was won after Siboniso Gaxa's fierce shot forced Jackson Mabokgwane into a great fingertip save.
Substitute Sibusiso Masina brought a goal back for AmaZayoni with 10 minutes remaining, tapping in at the far post from Lehlohonolo Nonyane's pinpoint delivery.
But it was not enough to spark a comeback as Steve Komphela's side held firm to put their name in the hat for the semi-final draw on Sunday afternoon.
Chiefs squandered numerous opportunities throughout the match and should have scored several goals, while Aces played their fair share of good football and hit the woodwork in the first half when Collins Mbesuma's strike crashed against Itumeleng Khune's crossbar.