The Scotland squad player scored either side of the break to sandwich Lee Wilkie's superb equaliser for the visitors in first-half stoppage time.
Derek Riordan was given a standing ovation when he made his second debut for the club as a 61st minute substitute but saw his thunder stolen by Fletcher.
Hibs ended the game with 10 men after another debutant, Sol Bamba, was shown his second yellow card in the second half for a late tackle on Jon Daly.
Hosts Hibs, inspired by a bumper home support, made an encouraging start and carved United open after just three minutes.
Dean Shiels fed Colin Nish inside the box but the towering marksman's pass was just behind strike partner Fletcher and the move fizzled out.
Northern Ireland international Shiels engineered Hibs' next chance in the seventh minute when he crossed for Fletcher but the striker headed just wide from six yards after out-jumping Danny Grainger.
Just when United were beginning to look comfortable, Hibs took the lead in the 21st minute through Fletcher after a great move.
Ian Murray's deep cross was knocked back to the Scotland Under-21 international with a cushioned volley from Nish and Fletcher's diving header beat Lukasz Zaluska from eight yards.
United were dealt a further blow when on-loan Sunderland striker Roy O'Donovan was forced to go off injured in the 27th minute and was replaced by Daly.
Despite the set-back, the visitors created their first chance of the game in the 31st minute.
Francesco Sandaza did well to create room on the edge of the box and played in Scott Robertson but the midfielder's attempted chip over the towering Yves Ma-Kalambay from a tight angle was easily gathered by the Belgian.
Bamba became the first player to be booked in the match when he slid in late on Robertson 10 minutes before the break, leaving referee David Somers with little choice.
United striker Sandaza came within inches of levelling just before the break after goalkeeper Ma-Kalamaby and defender Rob Jones left the ball to each other but the Spaniard missed the target after getting in between the pair.
The visitors did equaliser in first half stoppage time.
Defender Lee Wilkie's first effort from long range was blocked by Jones but the ball came back to him and he unleashed an unstoppable first-time shot from 25 yards that found the corner of the net.
Both sides contributed to a scrappy start to the second period as they wrestled for control of the match.
Craig Conway's diagonal cross found Daly unmarked at the back post but his tame header fell comfortably into the hands of Ma-Kalambay.
Seconds later, Shiels hit a left-footed shot just over from inside the box before Nish also missed the target with a long-range shot after linking up with Fletcher.
A massive roar engulfed Easter Road in the 61st minute as the Hibs fans were on their feet to welcome back Derek Riordan when he came off the bench to replace Joe Keenan.
Riordan, who rejoined from Celtic at the start of the month, played a part in Hibs restoring their lead just seconds after Ma-Kalambay made a great save to deny Sandaza from close range.
The marksman began the move with a pass to Shiels inside the centre circle and Nish was also involved before Fletcher placed the ball into the net from 18 yards with Zaluska out of position.
Hibs, however, were reduced to 10 men in the 68th minute when Bamba was shown his second yellow card for a clumsy challenge on Daly.
United once again stepped up the pressure after going behind but were met by a Hibs rearguard desperate not to relinquish their advantage for a second time.
Hibs goalkeeper Ma-Kalambay had to look alert as he raced from his goal to smoother the ball as Sandaza raced in.
Prince Buaben then missed the target in the 86th minute after the ball fell to him in the box from substitute Danny Swanson's corner.
Lee Wilkie missed an even better chance from inside the six yard box after he headed David Goodwillie's cross wide moments later.
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