SPL - Celtic return to SPL summit
First half goals by striker Gary Hooper and midfielder Victor Wanyama gave the home side a comfortable interval lead but United midfielder John Rankin pulled a goal back five minutes after the break.
It was an uneasy second half for Neil Lennon's side but they did enough to leapfrog the champions who had beaten St Johnstone 2-1 at McDiarmid Park earlier in the day.
In the eighth minute Hoops goalkeeper Fraser 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 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 SPL young player of the month award for December on Friday, rose highest in the United defence following a short corner between Georgios Samaras and Emilio Izaguirre to head the latter's cross past Pernis.
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.
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.
Forster had to make a great save from Daly's close-range header after United defender Paul Dixon got clear down the left to deliver a great cross but the Hoops held on in a tense finale.
Stokes was denied a goal in the final minute of the half by a sharp tip over from Pernis, after the Ireland striker had unleashed an effort from the right-hand side of the box after being set up by Scott Brown.
Celtic started the second half as brightly as they finished the first, with Pernis forced into a good stop from Samaras' left-footed drive from the edge of the box.
In the 50th minute, Dundee United gained a foothold in the game with a goal of real quality. With no-one closing him down, Rankin looked up and shot past Forster from 30 yards with a shot that clipped the post and went in.
It looked like Celtic were completely thrown by the goal, and they then wasted a good chance when skipper Brown and Stokes got in each other’s way as they closed in on a cutback down the left from Adam Matthews.
Three minutes before the hour, Forster did well to keep out a Daly header, after the Irishman had got on the end of a Paul Dixon cross from the left.
Samaras could have wrapped up the points in the 72nd minute with a diving header from Matthews’ cross, but it was directed straight at Pernis.
The same player blazed over in the last five minutes, but Celtic held on to regain top spot with 56 points from their 23 games, two points ahead of Rangers on 54 points from 23 games.