With two second-half goals by substitute Charlie Mulgrew putting Aberdeen ahead after Jan Vennegoor of Hesselink's early goal in the first half, it fell to substitute Scott McDonald to head home the equaliser.
This raised the temperature even higher inside the ground and in the dying seconds Vennegoor of Hesselink flicked home the winner.
It was a quiet opening to the match before Vennegoor of Hesselink shot home from ten yards after Georgios Samaras' powerful run had created mayhem in the Dons defence.
The visitors nearly equalised, but Zander Diamond's header was cleared off the line by Marc Crosas.
Aberdeen came more into the match before the interval and Artur Boruc did well to block Lee Miller's shot.
Aberdeen equalised early in the second half when Mulgrew swept home a Jared Hodgkiss cross after some superb build-up play down the right flank.
The visitors stunned Celtic even further when they went ahead. Miller was fouled in an aerial challenge with Glenn Loovens and Mulgrew curled his free-kick over the wall into the corner of the net.
Celtic sent substitutes Aiden McGeady and McDonald on as they sought to get back into the match and both were involved in Celtic's equaliser.
McGeady powered down the right before his cross found McDonald, who made no mistake with his close-range header at the back post.
Celtic then went flat out for the winner and McDonald's shot flashed narrowly over before Stephen McManus headed wide.
Just when it looked as though Aberdeen might hang on for a share of the points, McGeady's run ended with a cross which Vennegoor of Hesselink turned in at the near post.
There was still time for Darren Mackie to shoot wide when he was clean through with seconds remaining.