Striker Gary Hooper and midfielder James Forrest capitalised on slack early play by the visitors to give the Hoops a comfortable half-time lead, before manager Neil Lennon raged in frustration at the profligacy of his side who spurned numerous chances.
Midfielder Ki Sung-yueng even missed the target with a penalty after the break, before a goal by Pars striker Andy Barrowman with four minutes remaining saw the home side glad to hear the final whistle.
Celtic play St Mirren on Saturday and could close the gap further before Rangers travel to Kilmarnock on Sunday, but Lennon will have his players practising their shooting before then.
The Irishman's plans have been disrupted by injuries all season but he made two changes through choice rather than necessity for the visit of the Fifers.
Winger Kris Commons replaced Georgios Samaras, with midfielder Ki in for left-back Badr El-Kaddouri, as Joe Ledley moved back into defence to replace the Moroccan.
Dunfermline boss Jim McIntyre made three changes, with Andy Dowie, Gary Mason and Barrowman in for the injured Austin McCann, Paddy Boyle and Liam Buchanan, the latter two starting on the bench.
The Pars were architects of their own downfall, starting from when Hooper opened the scoring in the sixth minute.
Midfielder Paul Willis play a loose pass in-field from wide on the left which was picked up by the former Scunthorpe striker, and he raced clear of the stunned Pars defenders before slipping the ball past Paul Gallagher.
The second goal from Forrest came in the 13th minute after Pars defender Alex Keddie gave possession away in the middle of the park, allowing Celtic to break.
With the Dunfermline defence exposed Commons played in Anthony Stokes inside the box.
Although Gallagher saved his low drive, the ball broke to Hooper who cleverly set up Forrest to guide the ball into the net from eight yards.
It was looking like being a long night for the Fifers, as Celtic kept up the pressure.
Celtic captain Beram Kayal had a drive from 25 yards which Gallagher blocked with his feet, moments before Commons had an effort from further out which sailed over the bar.
The home side continued to pour forward and good work by Ledley and Stokes saw Gallagher somehow keep the ball out from the Irishman's cut-back, with the visiting defence in more disarray.
On the half-hour mark Forrest weaved his way past a clutch of Pars defenders before crossing to the back post for Stokes who, from an acute angle, could only send it back across the goal and the Fifers breathed again.
It appeared all too easy for the Hoops and moments later Forrest was set up by Stokes but watched his low drive from the edge of the box skip just past the far post, moments before Commons went equally close with a similar effort.
Two minutes from the break Hooper became the latest Celtic player to pass up on a decent chance when his first-time strike at the edge of the box from Forrest's cross went wide.
Paddy McCourt replaced Commons for the start of the second half and the sight of the talented Irishman must have concerned the Pars defenders.
But there was a low-key start to the second period, with Celtic over-elaborating at times as they swarmed almost constantly around the Dunfermline box.
Ki even missed a penalty in the 56th minute after Dunfermline defender Paul Burns brought down Forrest just inside the 18-yard box, according to referee Bobby Madden.
The South Korea international sent the spot-kick wide, to stunned silence.
Then, in the 62nd minute Ki set up Stokes inside the box. Although the striker's shot beat Gallagher it rebounded off the post and the bedraggled Pars once more got off lightly.
Dunfermline were making occasional forays forward, as they had in the first half.
In the 70th minute Hoops goalkeeper Fraser Forster was forced to concede a corner from a low drive by Pars substitute Joe Cardle, on for Willis, before making another save from another Dunfermline substitute Liam Buchanan, who had replaced Andy Kirk.
Samaras came on for Stokes with 10 minutes remaining but in the 86th minute Barrowman fired an angled drive from the edge of the box low past Forster and in at the far post to set up a nervy ending, which Celtic were glad to see out.
