In the 21st minute, with what was their first real attack, the United midfielder met a Paul Dixon cross to bullet a header past Hoops keeper Fraser Forster to fire the Terrors into next season's Europa League.
A much-changed Celtic side huffed and puffed for most of the game and Anthony Stokes passed up their best chance seconds before the interval when he missed the ball altogether yards from goal.
The champions might also have had a penalty late in the game when Emilio Izaguirre's cross hit the hand of United substitute Keith Watson but the Parkhead men were not at their best and any time they did get near the United goal they found keeper Dusan Pernis in good form.
Neil Lennon originally made seven changes, bringing back Forster, Mikael Lustig, Glenn Loovens, Charlie Mulgrew, Adam Matthews, Scott Brown and Gary Hooper to the side which beat St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 night.
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.
United boss Peter Houston made two changes to the starting 11 which lost 5-0 at Rangers on Wednesday, defender Gavin Gunning replacing Garry Kenneth and former Sligo Rovers player Richie Ryan making his first start in place of Johnny Russell.
It was a tepid start to the game and indeed, nothing of any real note happened until the home side took the lead in the 21st minute from a short corner between wide-man Gary Mackay-Steven and left-back Paul Dixon.
When Dixon's whipped-in cross came into the box midfielder Robertson leapt to bullet a downward header past Forster.
The rather straightforward opener for United cajoled Celtic into moving up a gear.
In the 29th minute Kris Commons' corner from the right was met by Rogne but when the ball dropped to Hooper, his close-range shot was blocked by United keeper Pernis.
Four minutes later a dangerous cross into the United box by Izaguirre, again one of Celtic's best players, saw Tannadice stopper Sean Dillon take a fresh air swipe at the ball, happy to see it go past for a goal kick.
Moments later, amid heavy rain, after Pernis had spilled a long-range effort from Hooper, Dillon's contact was true, as he swept the loose ball away from danger.
In the final seconds before the break Commons had a good shot from 25 yards parried high into the air by Pernis before the Scotland midfielder's cross from the left moments later was missed altogether by Anthony Stokes a few yards out.
Stokes' miss was enough to see him hooked at the interval, replaced by teenager Tony Watt, who had scored twice in his debut at Motherwell a fortnight ago.
Celtic forced a couple of corners in quick succession as the pressed for the equaliser but Commons' second from the left was met by defender Charlie Mulgrew who drove high over the bar on the turn from six yards.
In the 51st minute Lustig's cross from the right found the head of Watt but his effort was straight at Pernis, who made another save from the young Celtic striker's low drive moments later.
In the 56th minute, late replacement Rogne limped off to be replaced by Victor Wanyama, who was soon booked for a hefty aerial challenge on Tannadice midfielder John Rankin, in the build-up to a United move which ended with Mackay-Steven flashing a shot over the bar.
In the 64th minute, Dixon whipped in another good cross from the left which skipper Jon Daly volleyed towards goal only to see it blocked by Mulgrew, before a good challenge by Wanyama prevented Rankin getting a clean strike on the loose ball which went wide. United replaced Robbie Neilson with Watson and Mackay-Steven with Russell, while Dylan McGeouch replaced Brown.
Wanyama was perhaps lucky to escape a second booking after a foul on Richie in the middle of the park while, in the 75th minute, the home side might have had a penalty awarded against them by referee Iain Brines when Izaguirre's cross struck the hand of Watson inside the Tannadice box.
As Celtic kept up the pressure, Commons combined well with Hooper at the edge of the box but Celtic's top scorer drew his angled drive inches past the far post.
With two minutes remaining Mulgrew had to concede a corner to prevent Russell's shot testing Forster, following a United counter-attack, but the set-piece came to nothing and eventually Celtic ran out of time.