First half goals from Anthony Stokes and Victor Wanyama, plus a second half strike from the impressive Charlie Mulgrew, saw the visitors win comfortably at East End Park.
Second-placed Rangers also won, beating Motherwell 3-0 at Ibrox thanks to goals from David Healy, Sone Aluko and an own goal from Stephen Craigan.
As both teams picked up the three points today, Celtic stay top two points ahead of Ally McCoist's Rangers at the top of the league.
The deadlock was broken at East End Park courtesy of a fabulous effort by Stokes. The former Arsenal man picked up a loose ball on the edge of the Dunfermline box, took one touch to control it and another to curl it beyond Chris Smith and into the top left-hand corner.
Chances then ebbed and flowed for the visitors as both Gary Hooper and Stokes had efforts sail narrowly wide. While Dunfermline - without a home win all season - struggled to create a clear opening of their own in the first 45 minutes.
With the hosts content to go in at half-time just the one goal down, Celtic pushed for a second late into the opening half and were soon rewarded for their efforts.
Wanyama made a darting run into The Pars' penalty area and volleyed Charlie Mulgrew's free-kick emphatically beyond the stranded keeper to give Celtic a deserved two-goal lead going in at half-time.
It was much of the same after the interval, as Neil Lennon encouraged his side to add further goals to their first half lead.
Hooper first had a clever near-post header fizz narrowly wide. Then James Forrest drilled in a dangerous low cross after tricking his way past two challenges, but there was no Celtic player in sight to get on the end of it.
Then, halfway through the second half, and away side grabbed an audacious third. Mulgrew stepped up to take another set-piece, but while everyone was expecting a cross, the 25-year-old saw his shot dip high beyond the keeper.
The win in the end was comfortable and leaves Dunfermline still searching for a win against Celtic at home since 1990.
At Ibrox, a goal 10 minutes from half-time eased earlier concerns for the hosts as Rangers cannot afford to drop further points if they want to challenge Celtic for the title.
Right before the opening goal, moans were heard inside the ground as David Healy wasted a golden opportunity to break the deadlock but could only steer his effort wide.
But Healy atoned for his earlier miss, to side-foot home after a neat cut-back by Sasa Papac.
The second half was more straightforward for the home side and it wasn't long before they doubled the lead from an unlikely source.
Aluko had been impressive throughout and when the winger picked up a Steven Davis pass, the 22-year-old finished with great aplomb with his left foot to pick up his first for the club.
The home side then took the handbrake off and put plenty of pressure on a disappointing Motherwell defence. Then with almost 15 minutes to the clock, A Lee Wallace shot was cruelly knocked in by defender Craigan past his own keeper to put the result beyond doubt.
In the other fixtures in the SPL, Aberdeen won 2-1 at Dundee United thanks to second half strikes from Mohamed Chalali and Karl Arnason despite Jon Daly putting the hosts in front.
At Rugby Park, Kilmarnock also came from behind to beat St Mirren 2-1.
Lee Mair scored the opener to give the visitors the perfect start after ten minutes, but a goal either side of the break from Paul Heffernan and Danny Racchi, saw the hosts take all three points and leapfrog St Mirren up into seventh.