The Hoops warmed up for their Champions League trip to Borussia Dortmund on Tuesday in style with a dominant display in Perth.
Celtic have now scored 20 goals and conceded none in winning their opening six games of the league season.
After Daizen Maeda had an early strike ruled out by a VAR review, the visitors killed the game off with three goals in 10 minutes just before half-time.
Kyogo Furuhashi ended an 11-month drought without an away league goal to open the scoring and quickly had a second when he headed in Greg Taylor's cross.
In between his goals, the Japan international turned provider for Paulo Bernardo to curl home from the edge of the box.
Brendan Rodgers' men showed no mercy after the break to Saints, who sacked former Scotland boss Craig Levein earlier this month.
Callum McGregor fired in his fourth goal of the season from outside the box.
Maeda headed in the fifth before Adam Idah rounded off the scoring after being teed up by an outrageous backheel from Alex Valle.
"All the noise has been about Dortmund but I wanted a Champions League mentality here," said Rodgers.
"I thought the variety of our game was very good. Everything you’d want from your team."
Aberdeen made it 12 wins from 12 games in all competitions under Jimmy Thelin at Dens Park.
Topi Keskinen teed up Kevin Nisbet to open the scoring before Keskinen fired in the second himself to give the visitors a perfect start.
Simon Murray pulled one back from the penalty spot but Aberdeen held out to become the first side outside of Celtic and Rangers to win their first six Scottish Premiership games since they did it in the 2015/16 season.
"It’s still September and the league stops in May, so we need to be humble, keep working and push ourselves to be better," said Thelin.
Rangers are now eight points adrift of the top two but have a game in hand at home to Hibernian on Sunday.
Bottom-of-the-table Hearts needed a stoppage time equaliser from Scotland striker Lawrence Shankland to prevent a ninth consecutive defeat in a 1-1 draw against Ross County.
A 99th minute penalty from Ross Graham earned Dundee United a 3-3 draw at Kilmarnock.
Teenager Lennon Miller scored both goals as Motherwell moved up to fourth with a 2-1 win over St. Mirren in a match where both sides were reduced to 10 men.
