United struggled to make the most of their possession in a drab first half while Killie only threatened from set-pieces.
But the introduction of Collin Samuel brought United to life and a piercing run from the forward set up Hunt's deserved winner with 19 minutes left.
Levein - in the dug-out after serving a touchline ban during his debut win over Rangers last week - brought back Hunt from suspension and Stuart Duff, with Greg Cameron and Samuel dropping to the bench.
Killie were missing Allan Johnston, with Rhian Dodds stepping in.
United enjoyed the better of the opening half but neither side provided enough support to their lone central strikers.
The hosts dealt with a couple of Peter Leven long throws in a scrappy opening that was a sign of things to come.
At the other end, a Duff volley bounced wide and Hunt turned well to hook a volley just past from the edge of the box.
The Irish striker got the first effort on target in the 19th minute when he rose well to meet Barry Robson's cross, but could not get enough power on his header.
Another high ball fell for Robson just inside the box but slipped as he and Hunt went to shoot and Graeme Smith smothered the United skipper's toe-poke.
United were using the ball better than their visitors and Craig Conway's cross after good work from Mark Kerr just cleared the far post.
Killie defender Frazer Wright almost grabbed his third goal this week when he headed down from a Dodds free-kick, but Derek Stillie pushed it wide at full stretch.
Danny Invincibile burst down the right and fired wide within 10 seconds of the restart and the visitors continued to press.
Dodds sneaked in front of Stillie to collect Grant Murray's header but the keeper managed to turn it behind.
Invincibile and Steven Naismith were playing much closer to Colin Nish, but Levein replaced winger Conway with Samuel to immediate effect.
The Trinidad & Tobago frontman beat three men as he cut inside from the corner flag but Garry Kenneth mis-hit his cutback before Cristian Kalvenes blazed over from 18 yards.
United were far more positive now with Robson pressing down the left and Samuel putting some dangerous balls into the box.
And Samuel set up the brilliant winner when he raced away from Garry Hay and crossed low for Hunt, who fired a first-time half-volley into the top corner from 15 yards.
Nish turned and struck from 25 yards as Killie tried to hit back but the ball curled away from goal.
Gordon Greer headed just over his own bar after Robson's corner and the Killie defender soon had a half chance at the other end but his shot bounced harmlessly wide.
Robson cut in from the right but could not finish the game off as his left-footed strike curled away from the far post.
