Mulgrew's delivery led to a glorious curling opener from Anthony Stokes and Victor Wanyama volleyed home the defender's delivery five minutes before the break.
Mulgrew wrapped up a comfortable win against the bottom side - Celtic's 10th league victory in succession - when he curled home a 30-yard free-kick midway through the second half.
Celtic were also boosted by the return off the bench of Emilio Izaguirre, making his first appearance since breaking his ankle at Aberdeen in August.
Stokes came in for midfielder Beram Kayal, who has been ruled out for at least a month with the ankle injury he sustained during Celtic's 1-0 win over Rangers, which sent them top of the SPL.
Dunfermline brought in John Potter and Gary Mason for Joe Cardle and Jason Thomson, who returned to Hearts after his loan spell ended.
They spent most of the opening stages inside their own half but they were getting tight on their opponents in their defensive third.
The hosts scrambled away two corners from Mulgrew and survived a low cross across the face of goal from Georgios Samaras, who was playing on the left of a four-man midfield.
But they could do little about the 18th-minute opener. The home defence cleared Mulgrew's cross after the Celtic defender had worked a short corner with James Forrest and Stokes brought the ball down right on the edge of the box.
The Irishman was faced with a crowded penalty area but he picked his spot in the top right corner and perfectly executed a curling right-footed shot just out of reach of Chris Smith's hand and just inside the frame of the goal.
Smith then pushed over Wanyama's powerful 25-yard strike.
Dunfermline attempted to provide more support to lone striker Andy Barrowman but they struggled to make headway.
Their first chance fell to Martin Hardie after some persistent play from David Graham set him up 25 yards out but the veteran midfielder placed his shot two yards wide.
Celtic soon moved two ahead in the 40th minute following another Mulgrew set-piece.
The former Aberdeen player curled a free-kick towards the back post after Martin Hardie had fouled Samaras 35 yards out and Wanyama ran round the back to cushion a volley home from six yards.
Dunfermline almost responded immediately when they worked space for Paul Burns in the left channel but Fraser Forster narrowed the angle and stood up to beat away the midfielder's shot.
The more attack-minded Cardle replaced Mason in the 52nd minute and Dunfermline made a bright start to the second half.
However, Celtic stepped up a gear and Gary Hooper glanced Joe Ledley's cross beyond the far post.
Another chance went begging when Forrest played the ball across the face of goal after a solo run.
Dunfermline brought on striker Andy Kirk for Paul Willis in the 65th minute and Barrowman almost broke through after a long ball but Celtic got enough men to snuff out the initial danger and Hardie shot wide from 22 yards.
Celtic moved further ahead in the 69th minute when Mulgrew curled a free-kick which dipped over Smith's head and sailed into the middle of the goal.
Smith looked more assured when he parried a Samaras strike after Scott Brown had played the Greek striker in.
The goalkeeper then tipped over Forrest's 25-yard strike before Izaguirre came on in the 79th minute for his first appearance since August to a huge ovation.
Substitute Ki Sung-yueng volleyed just wide following a corner as Celtic cruised to victory.