Kris Commons gave the champions the lead with a 30-yard free-kick in the 11th minute before Georgios Samaras helped himself to a double, the first with a deflected drive in the 17th minute and the second a header 10 minutes later.
Neil Lennon's side retained their grip on proceedings in the second half, but only added to their lead with five minutes remaining when substitute Anthony Stokes, on for Commons, made it 4-0, also with the help of a deflection.
It was a perfect warm-up for the Hoops ahead of next week's William Hill Scottish Cup final against Hibernian at Hampden where they will look to clinch the double.
For United boss Jackie McNamara it was confirmation of the problems he will have to solve for next season, most of which, on this evidence, lie in defence.
The Celtic supporters started the match dishing out stick to United captain Jon Daly, who is set to join Rangers next season, and it appeared to unsettle the Irishman, who was playing in central defence.
In the seventh minute he clattered into Commons outside the area and when the free-kick was whipped in by the same Hoops player the ball came off Daly and almost spun past United keeper Radoslaw Cierzniak.
A few minutes later Daly again brought Commons crashing to the ground 30 yards from goal, in a more central position, and this time the former Nottingham Forest and Derby player curled the free-kick low around the wall and past the flailing Cierzniak.
United were contributing to their own problems.
In the 14th minute Barry Douglas was penalised for bring down Paddy McCourt 30 yards from goal when there was no imminent danger and Commons was on target again with another powerful free-kick.
However, this time Cierzniak tipped the ball on to the post before the home defence cleared it to safety.
However, the second goal was only moments away and came via Samaras's shot from inside the box which took a marked deflection off United defender John Souttar before looping over the helpless Tannadice keeper.
Gary Hooper and Commons had attempts from long range which flew wide of the target, but in the 26th minute Celtic keeper Fraser Forster had to rush off his line to block at the feet of Rory Boulding before Mark Millar tested the Englishman with a decent drive.
However, that was about the sum total of the home side's first-half resistance.
A minute later, after midfielder Millar had conceded a foul on Hooper wide on the right, Celtic scored again when Samaras leapt high inside the packed United six-yard box to head in Commons' whipped-in free-kick.
Celtic eased their way to the break and into the second half.
Hooper had the ball in the net from a yard out when he got on the end of a Samaras cross eight minutes after the break only to see the flag up for offside.
The tempo dropped by the minute but on the hour mark Hoops midfielder Joe Ledley bulleted in a header from a Commons corner and watched as Cierzniak flew to his top left-hand corner to pull of a fantastic save.
In the 68th minute his Parkhead counterpart was called into action for the first time in the second half when he made a good save from Daly's volley from 12 yards.
Celtic were firmly in control, though, with Hooper heading over from a Samaras cross and Cierzniak making another fine save, this time from Celtic skipper Scott Brown, whose volley almost sneaked inside a post.
There was time for Lennon to give a debut to Swedish-born Barhudin Atajic, who replaced Samaras, before Stokes made it 4-0 with a drive which took a touch off Millar to beat the hapless Cierzniak.