After Stephen McManus put the hosts ahead on the half hour mark, there was little chance of an upset, though with such a slender advantage there was always the possibility of one mistake changing the result.
It was perhaps understandable that Celtic looked jaded after Wednesday's Champions League heroics against Manchester United, but they still managed to squander enough chances to win several games.
Lee Naylor nearly caught out the Pars keeper Dorus de Vries with a snapshot and Jan Vennegoor of Hesselink soon had the keeper scrambling a couple of headers.
The visitors' keeper spilled a Thomas Gravesen shot, but Phil McGuire hooked clear and at the other end Gary Mason's swerving shot troubled Artur Boruc, but Celtic made the breakthrough after Souleymane Bamba conceded a corner.
Shunsuke Nakamura's inswinging cross was met flush on his forehead by McManus, who powered his header home past the stranded De Vries. Stephen Simmons looked guilty leaving the defender unmarked as he headed home.
The visitors nearly fell further behind when Gravesen unleashed a 25-yard strike which struck the post before the Dutch keeper had moved.
De Vries then did well to block Nakamura's effort with his feet after Shaun Maloney's pass had put the Japan international schemer through.
Celtic continued to press in the second half, but the tempo dropped even further with Dunfermline unable to take advantage.
Kenny Miller was again unable to break his scoring duck since joining the Bhoys on a free transfer from Wolves in the summer and only Neil Lennon and Gravesen efforts brought the crowd to life.