Neil Lennon 's men fell behind to an early Lee Erwin strike, but Shane Duffy headed home from a free-kick to get them back on terms.
James Forrest got what proved to be the winner before the half was out, but while the visitors dominated possession they couldn't find a third to put the game to bed.
Odsonne Edouard spurned an opportunity from the spot but it didn't prove costly as the champions won the first of their two games in hand.Lennon had opted to make changes, with Patryk Klimala replacing Albian Ajeti up-front while Hatem Elhamed came in for Christopher Jullien at the other end of the park.Celtic were in imperious form as they dispatched Ross County 5-0 at the weekend, but it was the hosts who took a shock early lead.
Read more on dailyrecord.co.uk