The Hoops were knocked out of the Scottish Communities League Cup semi-final by St Mirren at Hampden last Sunday but there was little chance of another shock, although the breakthrough did not come until the 56th minute when Commons scored from the spot after he was fouled by Raith defender Simon Mensing.
Irn-Bru First Division side Rovers ran out of steam and 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.
With forward Georgios Samaras and left-back Emilio Izaguirre out injured, Watt and Joe Ledley were handed starting berths by manager Neil Lennon, as was midfielder Beram Kayal with skipper Scott Brown dropping to the bench.
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, as the Raith defence continued to repel constant Celtic pressure which brought another fruitless penalty claim when Adam Matthews' cross struck Laurie Ellis, 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 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.
As Celtic continued to buzz around the home side's penalty area, Wanyama flicked a Matthews shot from outside towards goal but McGurn saved comfortably.
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.
Moments later at the other end, Matthews' right-footed drive from outside the box cleared the Raith bar only by inches, and when the right-back turned provider with a cross on to Watt's head, the Celtic striker's effort went just wide of the far post.
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.
Brown replaced Kayal and Forrest came on for Watt and Celtic dominated again.
In the 68th minute McGurn made a good save from Gary Hooper after the Englishmen had escaped a couple of tackles inside the Raith box, and although the ball fell to Brown he flashed his shot wide of the target.
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, who had missed a penalty in the defeat by St Mirren, curled in a third with four minutes remaining to ease the Parkhead side into the quarter-finals.