The Hoops went ahead in the 56th minute when Commons scored from the spot after he was fouled by Raith defender Simon Mensing. In the 82nd minute Forrest, on for Tony Watt, grabbed the second goal, with Mulgrew's fine strike four minutes later completing a solid afternoon's work by the visitors.
Celtic ignored the swirling wind to take control from the start. There was a shout for a penalty in the sixth minute when Mulgrew's cross was blocked by Raith defender Eddie Malone, using his arm according to the visitors, but referee Steven McLean was not convinced.
In the 21st minute, midfielder Victor Wanyama tested David McGurn with a drive from 25 yards which the goalkeeper saved with some ease.
On the half-hour mark Raith, in a rare attack, had a penalty claim of their own turned down when back-tracking midfielder Joe Ledley nudged Jason Thomson in the box to end a surging run by the Kirkcaldy defender. Again referee McLean was unimpressed.
Celtic retained a tight grip on the game and in the 35th minute Mulgrew's inswinging corner from the right escaped everyone before clipping the far post and running away from goal.
Raith started the second half in a more positive fashion with midfielder Allan Walker's cross from the right just evading on-rushing striker Brian Graham.
In the 53rd minute, after Kayal had tripped Malone 25 yards from goal, Celtic goalkeeper Lukasz Zaluska saved Graham's curling free-kick down at his left-hand post with defender Kelvin Wilson completing the clearance.
Two minutes later Celtic made the breakthrough when Mensing clumsily tripped Commons inside the box and the Scotland midfielder slammed the spot-kick low into the net.
The goal sparked Raith into life and for the first time in the game they threw everything at the visitors, albeit for only a few minutes and without any joy.
The game stretched further in the final stages and when Celtic broke with pace Forrest's first touch from Hooper's pass took him into the Raith box and he confidently knocked the ball past McGurn.
Mulgrew curled in a third with four minutes remaining to ease the Parkhead side into the quarter-finals.