The Dons rode their luck at times, particularly in the early part of the second half as the visitors pushed hard for an equaliser. However, Aberdeen were clinical in front of goal and ran out deserved winners.
Celtic will play a lot worse and win. But despite all their possession they found it hard to break down a resolute Aberdeen defence that was brilliantly marshalled by skipper Scott Severin.
The Bhoys were forced to make a late change to their starting line-up when first-choice goalkeeper Arthur Boruc sustained a groin injury in the warm-up and was replaced by Mark Brown.
Celtic were first to show and, after five minutes, Scott Brown raced onto a Barry Robson throughball and turned away from Mark Kerr before seeing his effort blocked by Severin.
From the resulting corner kick Gary McDonald mis-controlled the ball inside the area and it fell to Scott McDonald whose snap-shot was headed off the line by Richard Foster.
Minutes later at the other end a poor headed clearance by Gary Caldwell fell to Darren Mackie who played the ball into the path of Gary McDonald.
He appeared to be held back by Stephen McManus but, despite loud appeals from the home fans, the referee waved play on.
Sone Aluko was alert when he dragged the ball away from Andreas Hinkel, before driving in a great near post ball which Mark Brown saved at the feet of Lee Miller.
Aberdeen had taken a grip on the game and forced the champions onto the back foot.
The closest they came to breaking the deadlock in the opening quarter came in the 20th minute when Severin and Charlie Mulgrew combined for the latter to play great ball into the area where Mackie had nipped between the two central defenders.
However, the goal was disallowed as Mackie had stolen a yard and was correctly flagged for offside.
Aberdeen deservedly took the lead in the 24th minute. Miller played the ball wide to Stuart Duff, whose brilliant cross found Gary McDonald moving into the box and he easily nodded the ball home from six yards.
Celtic are not top of the table for nothing though and were back on terms within 60 seconds. Aiden McGeady burst past Kerr at pace before driving his cut-back into the path off Scott Brown who knocked the ball home from eight yards.
Aberdeen were back in front just before the half-hour mark. A poor clearance by Paul Hartley from Severin's free-kick fell to Kerr whose intelligent ball back across the area found Duff timing his run beyond Hinkel to gleefully smash a shot behind Mark Brown from 14 yards.
As the game approached half-time, a Shunsuke Nakamura corner was knocked just over the bar by McManus as Celtic went in search of a second equaliser.
However, it was The Dons that forced a great final chance of the half. Alexander Diamond got above McManus to head the ball into the path of Miller.
From the angle the striker clipped a delightful effort over Mark Brown only for Gary Caldwell to hook the ball away from under his own crossbar.
Celtic were first to show after the interval when McGeady broke away from his marker to play in Scott Brown only for Severin to make a crucial block at the expense of a corner.
The Bhoys were in total control at this stage and in the 56th minute McGeady waltzed past two Dons defenders only for Diamond to make a last-gasp block with Scott McDonald ready to pounce.
Four minutes later Hinkel's speculative cross-cum shot from wide on the right totally deceived Dons keeper Jamie Langfield before cannoning of the far upright. The ball fell into the path of Hartley who, under pressure from Kerr, blazed over the bar.
Celtic introduced Jan Vennegoor of Hesselink in the 68th minute and finally got the equaliser their second-half play had merited.
Robson twisted and teased Foster before playing in a tremendous cross which gave Scott McDonald the easiest of chances to nod home from four yards out.
Aberdeen were not to be denied and went ahead again on 76 minutes. Mulgrew's swerving free-kick looked to have gone beyond the far post only for Diamond to throw himself at the ball and guide his header just inside the post.
The Aberdeen fans were in raptures again two minutes later. Another Mulgrew free-kick caught the visitors defence out and it was that man again Diamond who easily knocked home from three yards.
Celtic went in search of a late consolation goal with Vennegoor of Hesselink climbing above Diamond to head the ball into the path of Scott McDonald whose 14-yard right-foot effort forced Langfield to make a brilliant one-handed save.