Melvin de Leeuw's eighth of the season gave County a surprise lead but Kris Commons equalised for the home side who could not go on to find a winner.
Anthony Stokes came close to converting Emilio Izaguirre's cross as Celtic made a bright start.
Stefan Johansen fired wide before De Leeuw (16) profited from Celtic's midfield uncertainty, latching onto Graham Carey's pass and drilling past Fraser Forster.
Carey and de Leeuw both threatened to add to the lead but when County keeper Mark Brown parried Leigh Griffiths shot, Commons (35) was first to the rebound to claim his 28th goal of the season.
Griffiths latched on to a Commons pass and only a last-ditch tackle from Erik Cikos prevented him from giving the home side the lead.
Celtic were now in the ascendancy as Liam Henderson's header come back off the post while Efe Ambrose and Izaguirre also went close.
Ambrose fired over from the angle of the box before Celtic made their first switch - Commons being replaced by Georgios Samaras, who was making his 250th appearance for the club.
But it was County who came closest to snatching a winner as Carey's drive struck the outside of the goalframe but the visitors were to hold out for an unlikely point in their push for top-flight survival.