First-half goals by striker Gary Hooper and midfielder Victor Wanyama gave the home side a comfortable interval lead and it looked like they would go onto coast the game.
However, visiting midfielder John Rankin pulled a goal back five minutes after the break to set up an uneasy second half for Neil Lennon's side.
In the end, though, Celtic did enough to leapfrog the champions who had beaten St Johnstone 2-1 at McDiarmid Park earlier in the day, while extending their winning run in the SPL to 11 games.
Hooper and Wanyama were two of five changes made by Lennon for the visit of the Taysiders.
Goalkeeper Fraser Forster, defenders Adam Matthews and Thomas Rogne also came in, with Lukasz Zaluska, Kelvin Wilson, Cha Du-ri, Ki Sung-yueng and Paddy McCourt all dropping to the bench.
For the Tannadice side, defender Robbie Neilson replaced Stuart Armstrong, who started as substitute.
United had not won in the east end of Glasgow since a 1-0 victory on Boxing Day, 1992 and they had manager Peter Houston in the stand, serving the first of a two-game ban for criticising match officials following their recent defeat by Aberdeen.
It was the visitors, though, who had an encouraging start to the game.
In the eighth minute Forster had to make a good save from United defender Garry Kenneth's drive from 12 yards after Gary Mackay-Steven's cross from the left found its way to the back post.
However, United found themselves two goals down in the space of five minutes.
In the 12th minute Hooper, subject of two rejected bids so far from Southampton, drove the home side into the lead after a swift Celtic break all too easily cut open the United defence following a mistake by Sean Dillon.
The Parkhead striker ran onto an incisive pass from Anthony Stokes before confidently sending a right-footed shot past Dusan Pernis from 14 yards.
Five minutes later Wanyama, who picked up his Clydesdale Bank Premier League young player of the month award for December yesterday, rose highest in the United defence following a short corner between Georgios Samaras and Emilio Izaguirre to head the latter's cross past Pernis.
The Tannadice side, with Mackay-Steven looking good, tried to work their way back into the game but it looked a long way back.
United were racking up the corners which hinted at danger but the Celtic defence stood firm.
The Parkhead men appeared to believe the points were already in the bag as they eased their way towards the break, albeit still well in control.
In the 41st minute a decent move involving Johnny Russell, John Rankin and Mackay-Stevens ended with skipper Jon Daly sending a drive from 25 yards just past Forster's left-hand post.
But moments later Rogne headed Charlie Mulgrew's inswinging corner from the right past the far post, before Pernis made a good save from Stokes' ferocious right-footed drive.
The second half began with Pernis fumbling a Mulgrew free-kick before the visiting keeper had a fine block from Samaras's left-footed drive from the edge of the box.
It looked like the afternoon might turn into a Celtic procession but in the 50th minute United found themselves back in the game when Rankin's long-distance drive sped past Forster and went in off the post.
The Tannadice midfielder's goal stunned the Hoops support who groaned moments later when skipper Scott Brown and Stokes got in the way of each other in the United box in their bid to latch onto Matthews' cut-back, after the Wales defender had worked hard down the right to reach the byline.
The Taysiders scented blood and in the 57th minute, Forster making a great save from Daly's close-range header after United defender Paul Dixon got clear down the left to deliver a great cross.
Celtic survived the consequent corner but nerves were being stretched.
Celtic needed the third goal to alleviate the growing tension and Samaras had the chance in the 72nd minute when he raced into the box to meet a flighted cross from Matthews but the Greece striker's diving header went straight to Pernis.
As the game entered the last 10 minutes United, sprightly and energetic, stepped up their quest for the equaliser but Forster was well protected by his rearguard.
Samaras fired over the bar with five minutes remaining and there was no real late push by United, to the relief, no doubt, of the home support.