The reigning champions were frustrated at home by Terry Butcher's Inverness side, who top the league with 10 points from their four matches.
Indeed, the leaders could feel aggrieved that they did not secure a fourth successive win as Celtic finished well to salvage a point.
|RESULTS IN FULL
An upset looked on the cards in the first half after Irish duo Aaron Doran and Richie Foran both netted. Defender Charlie Mulgrew pulled a goal back for Celtic just before the break though and it inspired the hosts in the second half.
The Bhoys – who were shocked 2-0 by Shakhter Karagandy in the first leg of their Champions League play-off on Tuesday – chased a result and eventually got their equaliser nine minutes from time. The source was another defender, as Welshman Adam Matthews collected a pass, cut inside and fired a low strike beyond Inverness goalkeeper Dean Brill and into the back of the net.
There would be no winner in store for Celtic but they could be content with a point given the precarious position they found themselves in during the first half. Neil Lennon's men remain unbeaten after three league matches with seven points.
St Johnstone may have finished third last season but they were thrashed 4-0 by a rampant Dundee United at Tannadice Park. Keith Watson's fourth-minute goal set his side on their way and striker David Goodwillie doubled their lead shortly after.
Winger Gary Mackay-Steven – reportedly a transfer target for Leeds United and Blackburn – added a third in the 39th minute before Stuart Armstrong wrapped up the rout just after the break.
Hearts' bid to reduce their 15-point deduction continued with a 2-1 home win over Aberdeen. The Edinburgh outfit – who are now eight points below zero – took the lead through Jamie Walker but were subdued by Kevin McHattie's red card and Niall McGinn's subsequent equaliser.
Substitute Jordan McGhee came off the bench to score a dramatic winner with two minutes to play, however, sending the Tynecastle crowd wild.
Other results on Saturday saw Liam Craig's second-half double help Hibernian to a 2-1 win at Kilmarnock, while Richard Brittain also netted twice as Ross County defeated St Mirren and Motherwell edged Partick Thistle 1-0 thanks to John Sutton's first-half strike.