The Hoops went into the game only needing a point, but did not come to life until after a dull, goalless first half when a Gary Hooper double and a strike by stand-in skipper Joe Ledley in between, got the celebrations fully under way.
Georgios Samaras, on for Anthony Stokes, completed the scoring for Celtic with a virtuoso goal two minutes from time as Aaron Doran netted an even later consolation for the visitors.
Celtic can now look forward to the William Hill Scottish Cup final against Hibernian at Hampden next month, where they will try to complete the domestic double. Combined with a run to the Champions League last 16, that would represent a fine campaign for Neil Lennon and his side.
Lennon watched from the stand, serving the first of a three-game ban after being found guilty of offensive language towards St Mirren skipper Jim Goodwin in a game at the end of March, and would have expected better in the first half in which side were a little sluggish.
Celtic got the breakthrough in the 60th minute when Hooper held off Graeme Shinnie before driving low past the helpless Reguero from 12 yards and then Ledley set up the overlapping Mikael Lustig before getting on the end of the return pass to guide the ball into the far corner.
Hooper's fancy flick from Kris Commons' driven cross from the left made it 3-0 in the 73rd minute and Samaras finished the game in fine style with a solo effort which ended with him crashing a shot from a tight angle in off the crossbar.
Doran's goal in the final seconds of injury time, as he capitalised on slack defending, was a mere footnote.