Chiefs needed extra-time against Tornado FC to get to this stage, and had won only two of their last five matches in all competitions.
Following their Soweto derby draw with Orlando Pirates, the Amakhosi made three changes that saw Bruce Bvuma, Ryan Moon and Siphelele Ntshangase coming in.
Chiefs give little space to the third-tier side, with George Maluleka missing the top corner by a whisker with a shot from outside the box on 13 minutes.
Moon was unfortunate not to get on the scoreboard as his effort came off the inside of the post after bringing down a long ball on 20 minutes.
The Magic held their own, with the visitors struggling to cause further problems thereafter, despite Daniel Cardoso sending over the crossbar in the latter stages of the half.
Chiefs ruffled the springs at the start of the second half, with the introduction of Khama Billiat for Hendrick Ekstein.
The poor finishing from Chiefs continued when Parker failed to volley home a perfect lay-off from striker partner Leonardo Castro on 63 minutes, while Magic had a response of their own at the other end, as Wade Crowie squandered an inviting opportunity with a dismal finish from inside the box.
Bearing the brunt of disgruntled supporters, Castro didn’t ease the situation when he missed an absolute sitter off Billiat’s through ball in the final minutes and the clash ultimately went to extra-time.
On the stroke of halftime in extra-time, a foul in the box on Billiat led to a penalty and Cardoso went straight down the throat of Cola to give Chiefs the lead.
Fellow substitute Philani Zulu double the lead on the 112th minute with a simple finish, before Billiat rounded off the scoring with a tap-in of his own to seal the Soweto giants' spot in the last eight.
Magic: Cola, Camb, Davids, Tchapi, Wood, Crowie, Jacobs, Mtsila, May, Warley, Mosa.
Subs: Van Neel, Dickson, Avontuur, Magoga, Mngomeni. Lucas, Maxbo.
Chiefs: Bvuma, Moleko, Hadebe, Cardoso, Mashiane, Maluleka, Ekstein, Ntshangase, Moon, Parker, Castro.
Subs: Vries, Ntiya-Ntiya, Mphahlele, Zuma, Zulu, Andrianarimanana, Billiat.