Forward Georgios Samaras gave the Hoops the lead with a well-taken volley after only five minutes with striker Anthony Stokes adding an even better second after 24 minutes as Neil Lennon's side dominated.
The home side rallied in the second-half but the Parkhead men always looked as they had enough in reserve if needed and on this form Celtic - who have only lost to Aberdeen in the league this season - could easily remain unbeaten in the five post-split fixtures.
With an eye towards next week's William Hill Scottish Cup clash with Rangers at Ibrox next weekend, United manager Jackie McNamara might be worried if his side replicate their insipid first-half performance although he will be happier with what they produced after the break even if it was not nearly enough to suggest they could even rescue a point.
Lennon kept his promise to tinker with his team selection in the final weeks of the season.
Midfielder Beram Kayal replaced Eoghan O'Connell with forward Samaras in for Leigh Griffiths and defender Steven Mouyokolo on the bench, available for the first time after recovering from an Achilles injury picked up last September.
United showed two changes with defender Andrew Robertson and striker Nadir Ciftci back from suspension as John Souttar and Gary Mackay-Steven dropped to the bench but it made little difference as the Parkhead side controlled most of the first half with some ease.
The visitors took the lead with the first effort on goal of the game after Tannadice striker Brian Graham had clipped the heels of Kayal, preventing the Israel midfielder burrowing towards the United box.
When Kris Commons curled the free-kick into the box Samaras won the race to the ball with a clutch of other players and volleyed into the roof of the net from 10 yards, with United keeper Radoslaw Cierzniak helpless.
The goal knocked the home side of their slumber and they began to make inroads into the game, with defender Gavin Gunning driving high over the bar from inside the box.
However, Celtic asserted themselves and their second goal came after a spell of good possession ended with Stokes playing a one-two with Commons and driving into the box past weak United attempts to intervene before slamming his shot high past Cierzniak.
It was all too comfortable for the Hoops who cruised to the interval, Stokes having another effort which Cierzniak.
Mackay-Steven replaced Graham for the start of the second-half which was only seconds old when United defender Keith Watson headed over the Celtic goal from 10 yards out.
Then Tannadice midfielder Ryan Dow ran on to a Mackay-Steven pass and knocked the ball past Celtic keeper Fraser Forster but the flag was up for offside.
Griffiths replaced Liam Henderson in the 52nd minute but it was the vastly-improved Taysiders who came close to reducing the deficit when Watson's left-footed shot from 14 yards slipped past the far post.
On the hour mark United skipper Sean Dillon failed to control a pass back from Dow, allowing Griffiths to pounce and fire in a low drive from 16 yards but Cierzniak pushed the ball past for a corner which the home side defended.
The former Wolves player had another effort from just outside the box moments later which escaped the far post, before Cierzniak denied Commons who had been slipped in by Virgil van Dijk.
It was another indication that Celtic were just as likely as United to get the next goal.
Forster had saves from Mackay-Steven and Stuart Armstrong in the final stages but Celtic were far from hanging on and the threee points they collected were well deserved.