Neil Lennon's team were confirmed as champions several weeks ago and, with nothing to play for, they were caught out by a United side who needed a positive result to seal qualification for next season's Europa League.
Midfielder Robertson decided the game when he met a Paul Dixon cross to bullet a header past Hoops keeper Fraser Forster midway through the first half.
A much-changed Celtic side struggled to find the momentum to recover and Anthony Stokes wasted their best chance seconds before the interval when he miscued a close-range effort.
Celtic thought they should have had a penalty late in the game when Emilio Izaguirre's cross hit the hand of United substitute Keith Watson, but referee Iain Brines correctly waved away their appeals.
Neil Lennon originally made seven changes, bringing back Forster, Mikael Lustig, Glenn Loovens, Charlie Mulgrew, Adam Matthews, Scott Brown and Gary Hooper to the side which beat St Johnstone 1-0 at Parkhead on Thursday.
However, Loovens had to withdraw after pulling up in the warm-up with a hamstring injury, his place taken by Norwegian stopper Thomas Rogne, with Cha Du-Ri being added to the bench.
It was a tepid start to the game until the home side took the lead in the 21st minute from a short corner between wide-man Gary Mackay-Steven and left-back Dixon.
When Dixon's whipped-in cross came into the box, Robertson leapt to bullet a downward header past Forster.
In the final seconds of the first half, Kris Commons had a good shot from 25 yards parried high into the air before the Celtic midfielder's cross from the left moments later was missed altogether by Stokes a few yards out.
In the 56th minute, Rogne limped off to be replaced by Victor Wanyama, who was soon booked for a hefty aerial challenge on Tangerines midfielder John Rankin in the build-up to a United move which ended with Mackay-Steven flashing a shot over the bar.
Dixon then whipped in another good cross from the left which skipper Jon Daly volleyed towards goal only to see it blocked by Mulgrew.
Referee Iain Brines turned down Celtic's penalty appeals in the 75th minute when Izaguirre's cross struck Watson's hand.
United are now guaranteed a place in Europe next season and their strong goal difference means they will almost certainly finish fourth.