Both goals came in the first five minutes of the game with Edward Ofere’s strike cancelling out Leigh Griffiths’ opener for the Hoops.
Celtic, chasing a fourth successive league win, were in front after only three minutes when Griffiths raced on to a long ball forward from Scott Brown and shrugged off the attentions of defender Gary Warren to fire a low left-footed shot into the bottom right-hand corner from 25 yards.
However, Inverness were level two minutes later when some slack defending allowed Ofere to blast home from close range after Jason Denayer had blocked Ryan Christie’s initial attempt on the line.
The hosts suffered a blow when goalkeeper Dean Brill was stretchered off and taken to hospital in the 20th minute after he had suffered a dislocated kneecap in a collision with Gary Mackay-Steven as he kicked clear.
His replacement Ryan Esson was tested by low shots from Stefan Johansen and Brown but he was equal to the task and he also made a fine save as he tipped Efe Ambrose’s powerful header over the crossbar.
Griffiths then managed to hit Row Z with a left-footed volley before directing a long-range free-kick high over the crossbar on the stroke of half-time.
Inverness made a positive start to the second half as Marley Watkins saw a header deflected over, with Josh Meekings spurning a chance with a close-range header from the resulting corner.
Esson made another fingertip save to keep out Denayer’s header from Stuart Armstrong’s free-kick at the other end, while Brown was booked for diving after tumbling in the area as he attempted to round David Raven.
Raven then flashed an angled drive over for Caley and Celtic substitute Kris Commons saw a long-distance volley sail over.
Greg Tansey also had a shot from the edge of the area deflected over for the hosts and Commons once again blazed over from long range, with the final chance of the match falling to Armstrong as his low drive was saved by Esson.
It was a fourth successive home draw for third-placed Inverness, whose winless run has now stretched to seven matches in the league, while Celtic failed to win after scoring first for the first time in 55 league matches.
The two teams meet again at Hampden Park next weekend in the semi-finals of the Scottish Cup, a game which can be seen live on Sky Sports 2 at 12.15pm on Sunday April 19.