Hibernian took the lead inside 40 seconds through Patrick Cregg but Jon Daly levelled from the spot in the 20th minute and Danny Swanson soon curled in a glorious second.
Peter Houston's side always had an extra edge and David Goodwillie and substitute Francisco Sandaza secured an impressive win after the interval.
Hibernian top scorer Anthony Stokes started on the bench and his injury time goal merely added a light sheen of respectability to the score.
United had recovered from an early red card for Paul Dixon to beat Motherwell 3-0 on Saturday and they faced another uphill struggle after Cregg quickly vindicated John Hughes' decision to hand him a rare start by netting from the first attack of the game.
The former Falkirk midfielder collected Ian Murray's driven cross and blasted a deflected right-footed shot high into the net from 16 yards.
Hibs continued to attack with menace down the left and almost went two ahead when Derek Riordan's corner fell three yards from goal but Dusan Pernis blocked from Chris Hogg.
Riordan's ambitious shot deflected a yard wide off team-mate John Rankin before United, with Morgaro Gomis back from injury, steadied the ship.
Garry Kenneth almost caught Hibs out with a quick free-kick but Hogg recovered to slide in on the recalled Daly.
Hibs threatened again when Murray was a yard underneath Rankin's inswinging corner, but Daly soon converted from the spot.
David Goodwillie's return pass evaded Danny Swanson but fell for Daly just inside the box before Hogg dived in to take the Irishman's legs without making contact with the ball. Referee Stevie O'Reilly's decision was unpopular with the home fans but it looked clear cut and Daly struck his penalty straight down the middle.
The visitors were ahead after Craig Conway's corner six minutes later when Swanson turned inside Liam Miller 15 yards out before curling a perfect left-footed shot into the far top corner.
United were looking comfortable as Prince Buaben pulled the strings in midfield and Goodwillie's movement put the home defenders on the back foot.
However, Colin Nish had a good chance to level just before the break when Rankin headed a loose ball beyond the visiting defence. The striker rounded Pernis but took the ball too far wide to shoot and his square ball was easily cleared.
Nish was played through again, this time by Miller, after a slick move seven minutes into the second half but he shot against the far post from 16 yards and Sean Dillon blocked Riordan's follow-up.
United also struck the post two minutes later when Daly evaded his marker to meet Swanson's corner on the volley.
There was little fluency to Hibernian's play and United moved further ahead on the hour mark when Goodwillie collected Gomis' pass and ran at Sol Bamba before unleashing a powerful 20-yard strike across Graham Stack.
Hibs stepped up the pressure with Riordan seeing a shot deflect wide before Andy Webster just did enough to stop Nish converting a low cross from Stokes, who had replaced Lewis Stevenson moments after United's third.
But Houston moved to a 4-5-1 formation to snuff out the home side and Sandaza slid home from 12 yards in the 88th minute after Goodwillie had led a counter-attack against an outnumbered home defence.
Stokes slotted home from eight yards following Riordan's flick but it did little to mute the visiting fans as their team stretched their unbeaten run to seven matches.