Goals in either half from Jamie Lindsay and Paul McMullan gave Stevie Frail and John Kennedy's kids the three points they needed to finish top of the table.
The Hoops went into the game on the back of a 2-0 derby win against Rangers on Tuesday, and knowing that only a win at Tannadice would be enough to keep the title at Paradise.
It was the home side who had the first chance of the game on six minutes when Gilmour headed over from McLaughlin's free-kick, but 10 minutes later, it was the Hoops who opened the scoring.
John Herron, playing at right-back, picked up the ball on the edge of the box and although his shot was blocked, midfielder Jamie Lindsay was on hand to fire home a powerful right-root shot from just inside the box that McGovern in the United goal could do nothing about.
And as the half drew to a close, the home side had a couple of chance to equalise. Leo Fasan in the Celtic goal denied Scott Smith with a superb save and then Malaga was close to turning in McLaughlin's cross at the far post in the final minute of the first half.
The second half began nervously, with neither side able to take control of the game, but it was Celtic who scored the vital second goal of the afternoon.
Liam Henderson gathered the ball 30 yards from goal and switched it out to the lively McMullan on the right flank. He raced towards goal, and from a tight angle just inside the box, fired a shot which hit the right-hand post and nestled in the back of the net.
Celtic, with the two-goal lead, looked to take control of the game, and while United pressed in search of a goal, chances were few and far between.
Indeed, the best chance came from Denny Johnstone in the last five minutes, which followed on from a Henderson header which, although on target, was well saved by the keeper.
Great work again from McMullan down the right flank saw him roll the ball along the 18-yard line, and Johnstone's powerful shot was tipped over the bar by McGovern.
CELTIC: Fasan, Herron, Fraser, F Twardzik, Waters, McManus, Lindsay, Henderson, McMullan (Hendry), Johnstone, P Twardzik. Subs not used: Hart, Donnelly, Hill, Tierney, McCart