John Hartson scored twice to give Celtic the points against 10-man Inverness Caley Thistle at Pittodrie.
The home side had Juanjo sent off early on after a clash with Neil Lennon and Hartson headed Celtic into the lead shortly afterwards.
Barry Wilson equalised from the penalty spot after Bobo Balde was penalised.
But Stilian Petrov gave Celtic the lead again with a fine volley moments after he missed a penalty, before Hartson scored the third from close range.
Caley Thistle started brightly, though it was Mark Brown who had to deal with a couple of Alan Thompson efforts from distance.
The home side's task was made a great deal harder in the 17th minute when Juanjo was red-carded for an off-the-ball clash with Lennon. The Celtic player escaped without any punishment, although both players appeared to be culpable.
The opening goal came after 25 minutes. Celtic strung a series of passes together before Didier Agathe crossed for Hartson to direct a header out of reach of the despairing dive of Brown and into the corner of the net.
Celtic were almost completely dominant by now and Brown thwarted Balde from close range and dived to his left to keep out an Henri Camara effort from 20 yards.
In a rare foray into the Celtic box, Caley Thistle should have had the equaliser after 41 minutes.
The ball fell to Wilson in plenty space and when his shot was blocked by Valgaeren, Liam Keogh scooped the ball over the bar with the goal at his mercy.
But the equaliser did come three minutes later. A corner into the box led to a scramble in the Celtic area and the grounded Balde was penalised for handball.
Wilson coolly slotted the penalty past David Marshall.
On the stroke of half time, Valgaeren should have put Celtic back in front but his header from only a few yards out was too high.
Celtic came even closer after 51 minutes when Hartson's cross was met by the head of Camara, but the ball came back off the crossbar.
Moments later, Hartson headed just wide and then Brown pulled off a tremendous save to stop the Welshman's drive from the edge of the box finding the top corner.
Petrov went from villain to hero within the space of a minute when, first, he had a penalty saved by Brown after Grant Munro handled in the area and then, from the subsequent corner, he volleyed past the keeper from the corner of the area.
The result was put beyond doubt in 76 minutes when the otherwise excellent Brown failed to hold a Valgaeren shot and Hartson was on hand to fire home.
Caley Thistle sub Bryan Prunty should have reduced the deficit but he failed to connect with the ball from only six yards out and Thompson curled a late free kick off the bar.