The home side put in a spirited performance but Celtic had a little too much class and despite an uncomfortable last ten minutes, it was Jan Vennegoor of Hesselink's 79th minute strike which secured a valuable three points for the visitors.
Dons boss Jimmy Calderwood included recent signing Lee Miller, who made his debut up front alongside Darren Mackie. He also recalled fit again Michael Hart and Gary Dempsey in an attempt to frustrate the visitors.
Despite his performances for Scotland during the past seven days, Kenny Miller could only find a place on the bench as the league leaders preferred the strike partnership of Maciej Zurawski and Vennegoor of Hesselink.
However, for the away fans all eyes were on Thomas Gravesen who Celtic signed from Real Madrid just before the transfer window closed. He took his place in midfield alongside skipper Neil Lennon.
The only incident of note during the opening exchanges was Gravesen getting a severe talking to by the referee for going in late on Chris Clark.
Against the pattern of play Aberdeen launched an attack on 12 minutes when Barry Nicholson swept the ball wide to Mackie who cut inside Lee Naylor before lifting the ball to the far post where Clark knocked his header back across the area only for Miller to fail to make contact with the goal gaping.
At the other end Gravesen's physical presence won the ball off Nicholson before releasing Aiden McGeady who advanced to the edge of the area from where he smashed a tremendous right-foot effort which was just wide of the upright with Jamie Langfield beaten.
Dons skipper Russell Anderson was a hero for the home side after 21 minutes when Richard Foster was caught in possession by Gravesen, but with the midfielder looking to cap his debut with a goal the Dons centre-back came from nowhere to make an absolutely brilliant block challenge.
Immediately after the restart Celtic were again in the ascendancy with Shunsuke Nakamura clipping the ball through to Zurawski who advanced into the box before screwing a shot wide from 10 yards.
At the other end, the pace of Mackie took him clear of Naylor but his cross just eluded Clark and flew harmlessly through for a goal kick.
A long ball out of defence by Artur Boruc got the Dons rearguard worried on 65 minutes when Vennegoor of Hesselink picked out Gravesen inside the area. His fierce first-time effort from 14 yards was brilliantly charged down by Langfield.
With 79 minutes on the clock Celtic finally got the breakthrough when Vennegoor of Hesselink's shot from 18 yards sneaked through a ruck of players and into the net beyond the unsighted Langfield.
Aberdeen looked to have got themselves back on level terms on 86 minutes when Anderson's header from a Dan Smith corner beat Boruc but was knocked on to the bar and off the line by Stephen McManus.