Hooper converted a Mark Wilson cross from close range in the 13th minute to put the visitors ahead but David Goodwillie drew the hosts level with a deflected shot just before the break.
Celtic pushed for the majority of the second half and there was some controversy when referee Dougie McDonald awarded them a penalty after United keeper Dusan Pernis had clashed with Hooper - then changed his mind to give a drop ball after consulting with assistant Steven Craven.
It looked like the match was heading for a draw but Neil Lennon's men were rewarded for their pressure with two minutes remaining when Hooper stabbed the ball in from close range, following a goalmouth scramble.
It was typically enthralling fixture between these two teams at Tannadice.
Celtic, skippered by Shaun Maloney in the absence of injured Scott Brown, carved open the first chance in the fifth minute when Hooper split the Tannadice defence to send Georgios Samaras clear.
However, the Greece striker declined to drive on at goal, instead sending an all-too casual volley from distance wide of the target.
Moments later, with the match still settling down, Anthony Stokes sent a right-footed past the post from the edge of the box.
The Hoops clearly had the edge and got themselves in front in rather easy fashion in the 13th minute when right-back Wilson's cross from the right was scooped in to the roof of the net by Hooper from a couple of yards out.
Stokes sent Samaras racing down the right and the former Manchester City striker confidently rounded Pernis but his angled-shot was cleared by United stopper Garry Kenneth before it sneaked in at the near post.
The home side hit back, forcing several corners and twice Samaras blocked attempts by Scott Severin getting through to Fraser Forster. At the other end Pernis spilled a Stokes drive after good work by Wilson again, and was happy to see the ball cleared to safety.
In the 28th minute Craig Conway got down the left and delivered a terrific cross to the back post but David Robertson just cleared the bar with his header.
The Celtic defence looking vulnerable at times and United midfielder Prince Buaben missed the target from 16 yards with a poor attempt at a volley.
In the 38th minute United were level. Joe Ledley was robbed in midfield by Buaben, allowing Goodwillie to drive from the edge of the box with the ball taking a marked deflection off Loovens to beat the helpless Forster, who had committed himself.
The game turned in United's favour but just before the break Pernis tipped a long-distance effort from Ledley over the bar and it was a frustrated travelling support which jeered the half time whistle.
United started the second half on the ascendancy, pushing the visitors back towards their own box and forcing a series of corners.
The Hoops emerged unscathed but only after Forster had produced a fine save in the 52nd minute from Buaben's 30-yard drive.
That let-off sparked life in the Parkhead side and in the 55th minute, after Conway had been penalised for a foul on Ledley 30 yards from goal, Maloney curled the free-kick just wide of the post.
The new Celtic skipper then had a decent shot saved by Pernis, after he had created a yard of space for himself inside the United box by skipping past Sean Dillon. The visiting fans were about to acclaim another Hooper goal when his shot from 14 yards beat Pernis but the ball skimmed the bar.
In the 69th minute Celtic appeared to have been awarded a penalty when referee McDonald pointed to the spot after Pernis dived at Hooper's feet to rob him of the ball.
However, after consulting his assistant Craven, McDonald gave a bounce ball in the box.
McDonald then had words with angry Celtic boss Lennon before his stand-side assistant Charlie Smith alerted the referee to missiles that had been thrown on the pitch.
As Celtic piled forward for the winner, Pernis brilliantly parried Stokes' shot from 12 yards before Ledley crashed the rebound off the post.
But with two minutes remaining, after the United defence had failed to clear properly a cross from substitute Paddy McCourt, on for Maloney seconds earlier, Hooper stabbed the ball in from close range for the dramatic winner and set up a mouthwatering clash between the Glasgow giants, both of whom have won their first eight SPL games.