Celtic were without five regulars, which gave youngsters Ross Wallace and John Kennedy the opportunity of rare starts.
However, it was not until the hour mark that the re-arranged home team could finally relax as they moved into a two-goal lead thanks to a Chris Sutton double.
Thistle, the only team in Britain without a League victory to their name this season, shocked Celtic with an early goal.
Kenny Milne forced his way forward from the halfway line only to be brought down by Kennedy at the edge of the box.
From the free-kick Derek Fleming's cross somehow found the smallest man on the park, James Grady, who headed home off the underside of the bar.
Celtic were quick to respond to this setback, but Henrik Larsson surprisingly headed wide a superb Wallace cross.
Sutton headed over before Larsson's powerful drive was well held by Kenny Arthur as the home team stepped up the pressure.
Celtic's equaliser had an element of good fortune about it. Momo Sylla's cross spun high off a defender before falling to Sutton who rolled the ball into the path of Larsson and Celtic's top striker finished in clinical fashion.
John Hartson's glancing header drifted inches wide before the home side took the lead.
After Wallace was fouled Larsson's free-kick was diverted home by Sutton, leaving Arthur helpless.
The visitors were still causing problems in the Celtic defence and wing-back Jean-Yves Anis advanced to test Magus Hedman with a powerful shot.
Early in the second half Wallace cut in from the left wing before unleashing a shot which Arthur did well to touch over the bar.
John Paul McBride had Hedman scrambling at his near post before Celtic scored their third goal.
It was Larsson who brilliantly controlled a long clearance before rolling the ball across the box for the advancing Sutton to slide home under keeper Arthur.