In the 33rd minute of one of the most one-sided games of the season at Parkhead on Wednesday night, Mulgrew - in Craig Levein's squad for the friendly against Slovenia next week - fired high past Chris Smith from 30 yards for the opener.
The Fifers goalkeeper saved the defensive-minded Pars from more suffering until the 76th minute when substitute James Forrest, on for Emilio Izaguirre, made it number two from close range.
It was Celtic's 13th consecutive win in all competitions, equalling a feat achieved by Martin O'Neill's side in 2004, and it confirmed, if needed, that Neil Lennon's men have the title all but sewn up, with a treble in their sights.
The Hoops manager made four changes with a view to utilising his squad in the run-in, as he had promised to do.
In came full-backs Cha Du-ri and Izaguirre and midfielders Ki Sung-yueng and Kris Commons with Adam Matthews and Forrest on the bench and Victor Wanyama and Thomas Rogne out altogether.
Dunfermline boss Jim McIntyre also made four changes, with defender Andy Dowie, midfielders Paul Burns and Ryan Thomson, and striker Liam Buchanan in for Martin Hardie, Gary Mason, Andy Kirk and Jordan McMillan.
The Fifers could have played them all and it would have been unlikely to have made a difference as Celtic dominated from start to finish.
It soon became clear the Pars would get 10 men behind the ball and leave Buchanan to forage on his own in attack.
Cha led the Celtic charge down the right flank but in the eighth minute it was his South Korea team mate, Ki, who tested Smith with a drive from distance which the keeper saved.
In the 19th minute, following the Pars' first corner of the game, Gary Hooper escaped the Dunfermline rearguard for the first time but Smith was adjudged to have got a touch to his shot to send it just past the post but again the Parkhead side failed to capitalise from the corner.
It was one-way traffic, Smith made another decent save, this time from Anthony Stokes' drive before Commons fired over the bar.
Lennon's side went down the right, down the left and through the middle in their bid to find space and Hooper succeeded again on the half-hour mark but his header was gathered by Smith.
But the breakthrough eventually came in the 33rd minute when Mulgrew decided he had had enough of the subtle approach and from 30 yards fired in a left-footed drive which gave Smith no chance.
As the home side set about sealing the win before the break, Hooper set up Commons but again Smith blocked the drive from 12 yards for a corner which came to nothing.
It was relentless pressure. Commons had yet another effort, this time a decent drive from outside the box, which Smith parried to safety.
In a fleeting appearance in the Hoops' box, defender Alex Keddie had a header from Joe Cardle's corner easily saved by Fraser Forster, who at last saw a bit of action.
The second half began in similar fashion to the way the first had ended with Celtic attacking and Dunfermline defending.
In the 50th minute Stokes had a great chance to make it 2-0 when Izaguirre's cross from wide on the left ended up at his feet eight yards out but with plenty of time and under no pressure, he slid the ball past the post.
Next it was Hooper's turn to miss a good chance, the former Scunthorpe striker screwing the ball wide of the target from 10 yards after the ball had broken to him when Cha had failed to get on to a clever pass from Commons.
There was simply no respite for the Pars and in the 55th minute Celtic midfielder Joe Ledley fired at Smith after Cha had set him up with a fine cut-back, before the Fifers keeper had to parry again from Hooper, who then thrashed the ball over the bar from close range to end another home attack.
Just after the hour mark Thomson's goal-bound header from a Paul Willis corner, his first touch since coming on for Cardle, was headed off the line by Izaguirre.
Smith was soon back in action and he did well to tip a shot from Celtic skipper Scott Brown over the bar after good work from Ki with nothing coming from the corner once more.
In the 76th minute, substitute Forrest stabbed in Hooper's cross at the back post for number two.
The turkey-shoot continued with Commons flashing a shot from the edge of the box over the bar before Smith fumbled his low drive from similar distance before the ball was eventually cleared to safety.
There was time for the luckless Smith to make another two saves from Hooper and the final whistle came as a relief to the battered visitors as Celtic march on.