The Hoops had struggled to break down a well organised Peterhead defence before the Irishman slotted the ball home eight minutes before the break on Sunday.
And Stokes drilled home a second after 57 minutes after linking up with Emilio Izaguirre before netting a third in the closing stages to end any hopes of a Blue Toon comeback.
Celtic made six changes, with Paddy McCourt replacing top scorer Gary Hooper who was out with a groin injury while Izaguirre also started following his return from a broken ankle.
Lukasz Zaluska was handed a rare outing in goal for the Hoops, while Cha Du-ri, Kelvin Wilson and Ki Sung-yueng were also drafted in, with James Forrest rested and Victor Wanyama and Adam Matthews dropping to the bench.
Peterhead boss Jim McInally opted for an attack-minded line-up with striker Rory McAllister leading the line supported by Martin Bavidge, while midfielder Jamie Redman passed a late fitness test to take his place in the side.
This was the first competitive match between the two sides but the part-timers looked far from over-awed in the opening minutes and were eager to test the Celtic rearguard.
However, it was the visitors who carved out the first real opening after 15 minutes when Georgios Samaras fed Stokes but Ryan Strachan made a vital block to deny the Irishman.
Celtic should have taken the lead after 27 minutes when Cha cut the ball back for Stokes at the near post but the Celtic striker blasted high over the crossbar.
McAllister picked up the first booking of the match a minute later for a late challenge on Mulgrew before Ki rattled the woodwork with a rasping drive from the edge of the area.
Peterhead continued to weather the relentless Celtic attacks, with Stokes once again was unable to find the target having been picked out by McCourt before Paul Jarvie did well to keep out a curling Mulgrew free-kick.
However, the opening goal arrived after 37 minutes when Samaras latched onto McCourt's diagonal pass and the Greek picked out the unmarked Stokes, who had the easiest of finishes.
The Irishman thought he had doubled the lead two minutes before the break when he netted from another Samaras cross but the goal was chalked off for offside.
Celtic had the ball in the net once more within three minutes of the restart after a neat finish by Stokes but again the striker had strayed into an offside position.
However the Irishman finally got his second after 57 minutes when he linked up superbly with Izaguirre before drilling the ball expertly into the bottom corner past the helpless Jarvie.
The visitors were dominating possession against a tiring Blue Toon, with Stokes curling a free-kick over the bar from the edge of the area as he searched for his hat-trick.
Scott Brown saw his powerful shot deflected over as Neil Lennon's side continued to press for a third.
Stokes duly completed his hat-trick seven minutes from time after Brown picked him out at the back post in plenty of space and the former Hibernian man made no mistake as he fired the ball past Jarvie to the delight of the noisy travelling support.