It took an extra-time goal from Lebohand Maboe (119') to seal a 2-1 come-from-behind win after Ime Okon's 36th minute goal had cancelled out a fifth minute opener from Therlo Moosa.
While SuperSport are enduring a difficult season in the South African top-flight in 12th place on the standings, Spurs are languishing near the foot of the Motsepe Foundation Championship, in 15th place.
SuperSport were pushed all the way by their first division opponents, leaving Hunt relieved to have won the game.
He said on SuperSport: "Good performance again, going a goal down and fighting.
"They got a lot of experience on their side, they got a lot of guys that played a lot of football, a lot of them played for me.
"They settled in and made it difficult for us. The turnover is obviously going to be important for them, and we handled it very well."
Hunt added: "There were one or two key moments that were scary, but we had a lot of opportunities around the box.
"We got to be a little bit better, but we got through; that's the main thing, we got a little bit of momentum."
SuperSport are next in action away to Kaizer Chiefs on Tuesday.
Kaizer Chiefs forward Wandile Duba is refusing to let a lack of playing time derail his ambitions, despite facing one of the most competitive attacking line-ups in the Premier Soccer League this season.
Durban City coach Gavin Hunt says it is only a matter of time before his team start to find their form in front of goal.
Terrence Mashego says he is enjoying his football again and has credited Gavin Hunt for giving him a chance to rediscover his form at Durban City.
Gavin Hunt is expecting a tough encounter when he faces former team Kaizer Chiefs in the Betway Premiership on Friday.
