Hibs snatched all three points thanks to a last-minute header from defender Gary Smith at Easter Road.
Just when it looked set for a draw, Smith headed in from close range after Dean Shiels had hooked the ball in.
Hibs lost skipper Ian Murray eight minutes from time when he was red-carded after an off-the-ball incident spotted by assistant referee Steve Pullar.
United were first to threaten, and Jim McIntyre did well to get a shot in on the turn at the edge of the area but Simon Brown was well positioned to save.
There was another escape for Hibs after 14 minutes when James Grady managed to get his head to a Barry Robson cross and Simon Brown lost his footing, but Gary Caldwell was on hand to hook the ball off the line.
United were well organised in defence with captain Paul Ritchie a towering presence, and Hibs were frustrated for long spells in the first half. Their first effort on goal did not come until the 20th minute when Dean Shiels poked the ball through for Derek Riordan, but he dragged his shot well wide.
But it was a signal for Hibs to finally show themselves as an attacking force towards half-time.
Garry O'Connor had a great opportunity to open the scoring when Scott Brown sent him clear, but he directed his shot straight at Tony Bullock. Riordan then tried to catch out Bullock with a chip from the edge of the area but the ball drifted over.
A goal was imminent and, in a flowing move, Riordan cracked a shot against Bullock's right-hand post from an O'Connor cutback.
Hibs finally managed to make the breakthrough 10 minutes from half-time. O'Connor fed Riordan on the right, and his cut-back picked out Shiels who rifled a 10-yard shot high into the net.
Hibs were irresistible at this point, and Glass struck the other post with a 20-yard shot after his initial effort had been blocked.
Riordan tested Bullock again with a 25-yard free-kick early in the second half which the goalkeeper plucked from underneath the crossbar. Jim McIntyre wasted a good chance at the other end when a superb first touch gave him a clear view of goal, but his finish was woeful.
The striker had another chance after Steven Fletcher lost possession, but his 20-yard shot was tame and Simon Brown saved easily.
Hibs did not heed the warning. After 63 minutes, McIntyre rose unchallenged to head home a Mark Wilson corner.
Hibs brought on Kevin Thomson midway through the second half for his first action since May last year, and within two minutes were ahead. Glass sent O'Connor clear and he drilled an unstoppable shot past Bullock from a tight angle.
United brought on Jason Scotland after 76 minutes, and within a minute he had equalised after accepting a Greg Cameron pass after Gary Caldwell was caught out.
But, after Murray's dismissal, Hibs dug in to get the victory with Smith's goal.