They had to play second fiddle to Dunfermline for much of the game but were clinical in front of goal and that proved to be the decisive factor.
Celtic were without Neil Lennon and Alan Thompson after their red cards at Ibrox the previous week and it was not the most fluent of performances by Gordon Strachan's team.
The Pars won two quick corners but any early promise evaporated as they gifted Celtic two goals in the first ten minutes.
In the fifth minute Mo Camara sent Zurawski scampering clear and the Dunfermline defence failed to close him down as he rifled a right-foot shot past Bryn Halliwell.
Halliwell was then badly at fault as he miskicked a clearance straight to John Hartson who was left with the simple job of side-footing the ball into the empty net.
The hosts lost Gary Mason with a cheekbone injury after 20 minutes but the Pars still outplayed Celtic for long spells.
They should have pulled a goal back after 26 minutes when Mark Burchill was fouled in the penalty area by the clumsy Adam Virgo. But Burchill's spot-kick was brilliantly saved by Artur Boruc, who pushed the ball away for a corner.
If Dunfermline had most of the first-half possession, Celtic were always more direct. Nakamura forced a fine save from Halliwell and Craig Beattie sent a shot into the side-netting as Celtic sought to kill off the game before the break.
The third goal was only delayed until the 57th minute when Zurawski picked out Nakamura and he had the simple task of stroking the ball past Halliwell.
Beattie fired narrowly over the crossbar after more good work from Zurawski as the heart went out of Dunfermline.
A mistake by Scott Thomson let in Zurawski for his second goal after 74 minutes to complete the rout.