Tannadice striker David Goodwillie had missed an early one-on-one with Celtic keeper Fraser Forster at Parkhead before Gary Hooper drove past Dusan Pernis in the 23rd minute for his 11th goal of the season.
It looked a long way back for United at the break but as the second half unfolded and Neil Lennon's side failed to get the clinching second, the Parkhead nerves began to jangle.
However it looked as if Celtic would hang on but with added time almost up, Dillon knocked in a floated free-kick from Barry Douglas to stun the home fans and keep Celtic three points behind Rangers at the top of the table.
It was a blow for the home side who, after watching Rangers win at Kilmarnock earlier in the day, were looking to keep right on their tails.
With striker Anthony Stokes out with a virus, Daryl Murphy returned to partner Hooper.
Midfielder Joe Ledley returned from suspension to replace Efrain Juarez who dropped to the bench.
United boss Peter Houston kept the same side which drew with Kilmarnock last week with Goodwillie leading the line.
The last meeting between the teams at Tannadice in October saw referee Dougie McDonald rescind a penalty award for Celtic which initiated a spat between the Parkhead club and the Scottish Football Association which had not yet been resolved by kick-off.
However, there was no such controversy in this game in which Celtic started strongly.
In the third minute Emilio Izaguirre's corner from the left was met by stopper Daniel Majstorovic whose wayward header arrived at Murphy almost on the byeline, and he headed off the post before the danger was cleared.
But Goodwillie missed a great chance two minutes later when Johnny Russell sent him racing through clear on goal with the Hoops defence missing.
The United striker, who made his Scotland debut as substitute in the 3-0 win over the Faroe Islands at Pittodrie in midweek, was too casual in his finish with his chip going past Forster's left-hand post.
The Taysiders looked dangerous going forward but in the 23rd minute Celtic got the breakthrough and there was no surprise that Hooper was the scorer.
The former Scunthorpe striker notched twice in the win against United in the controversial meeting last month and came up with a last-gasp winner at St Mirren last week.
After Tannadice skipper Garry Kenneth failed to clear his lines in the box the ball fell to Hoops captain Shaun Maloney who cut the ball back for Hooper, who, from 14 yards, turned and drove high in the net past Dusan Pernis.
Four minutes later Hooper should have made it two nil from four yards after being set up by the impressive Izaguirre but Pernis somehow managed to save the striker's volley on the line and the ball was escorted to safety.
It was all Celtic and in the 37th minute Pernis had to make another save from Hooper after the Englishman had latched on to Paul Dixon's poor pass to race in to the box, moments before Ledley blazed over the bar from the edge of the box.
Five minutes from the break, with the home side still on top, Celtic wide-man Niall McGinn came inside to fire a drive inches over the bar.
McGinn drove over the bar from the edge of the box two minutes after the restart as Celtic took control again.
United appeared sapped of real belief as the home side moved incessantly towards Pernis.
However, in the 52nd minute, with most of the Parkhead players camped around the United box, Goodwillie again broke free down the right but he failed to lift his head to see the unmarked Morgaro Gomis in support and was crowded out in the end, only winning a corner which came to nothing.
Things were rather quiet until 10 minutes later when Hooper took a McGinn pass and forced another good save from Pernis with a right-footed drive from inside the box, Celtic again failing to capitalise from the corner.
In the 71st minute, in a United breakaway, Goodwillie was bundled to the ground at the edge of the box by Majstorovic after he had tried to control Gomis's pass but referee Mike Tumilty ruled that there had been no infringement.
Kenneth missed the target twice with close-range headers as United pushed for the equaliser before Tannadice midfielder Prince Buaben cleared Ledley's shot off the line in a Celtic breakaway.
However, in the third minute of added time, Dillon dramatically headed in a Douglas free-kick from close range to give United a well-deserved point.