Trevor Mokwena opened the scoring with a brilliant strike early in the second half at the Richards Bay Stadium.
Knox Mutizwa levelled in the 84th minute, and substitute Mbulelo Nombengula fired home in extra time to put the hosts ahead.
Lundi Mahala then added a third goal with 120 minutes on the clock to seal Richards Bay's progression to the semi-finals.
Speaking after the game, Hunt admitted his side defended poorly, alongside wasting a number of chances in front of goal.
He said on IOL: "In the first half we were very good and we could've scored. Second half we score, but then with bad defending we met them back, and the second goal again bad defending.
"We also gave the ball away a lot today. Our one-on-one defend was poor, especially for the first goal. We have to stand up there and we can't let them cross. The scoreline is a bitter pill to swallow, but I'm very disappointed."
City are next in action in the Betway Premiership against Kaizer Chiefs at the Moses Mabhida Stadium on Friday.
"We just have to look at the next game. We know what we can do and what we can't do. We obviously need to look at a big game next week and we need to be better again," said the head coach.
