The Soweto giants kept their cup dream alive after overcoming ABC Motsepe League outfit The Magic FC in extra-time.
Chiefs will now face the men from the Mother City, who thrashed Highlands Park 3-0 in the round of 16.
Komphela will make a return to the Free State for the second time this year as his Golden Arrows side take on Bloemfontein Celtic. The two sides played to a 1-1 draw in the Absa Premiership last month.
The only National First Division sides left in the competition will do battle against each other, with Cape Umoya hosting TS Galaxy.
The Premier Soccer League will confirm the dates and venues for the encounters in due course.
Full quarter-final draw:
Bloemfontein Celtic vs Golden Arrows
Bidvest Wits vs Chippa United
Kaizer Chiefs vs Cape Town City
Cape Umoya United vs TS Galaxy