When the Bhoys took the lead through Kenny Miller after only four minutes it looked as though the home team were about to inflict a heavy defeat on the Pars.
However, missed chances meant it took until 15 minutes from time, when Jan Vennegoor of Hesselink glanced his header home, before Celtic looked assured of victory.
A great strike by Adam Hammill four minutes from the end led to an anxious final few minutes, but Celtic held on for a deserved victory.
Celtic's opener was instigated by Paul Telford's superb crossfield pass which Aiden McGeady gathered before outpacing Darren Young and squaring the ball to leave Miller with a simple tap-in.
Celtic then proceeded to squander chances. Miller's shot from a Shunsuke Nakamura pass lacked power and Craig Beattie's powerful header was superbly touched over by Dorus de Vries.
Nakamura shot wide from 15 yards and Miller was unable to divert Thomas Gravesen's shot on target.
At the start of the second half Miller saw a second goal ruled out for offside and Beattie miskicked in front of goal. Skilful play by Nakamura created another chance, but Miller lashed his shot against the crossbar.
The Dunfermline keeper made two fine saves from Lee Naylor and McGeady before Celtic scored their second.
Nakamura directed his corner to the edge of the area where Gravesen's volley was headed home with the finest of touches by Vennegoor of Hesselink.
The visitors pulled one back when Hammill gathered a throughball before creating some space and curling a superb 20-yard shot over Artur Boruc, but it was too little too late.