After midfielder Ki Sung-yueng had given the home side the interval lead at Parkhead on Saturday, substitute Paddy McCourt scored a wonderful solo second in the 65th minute which looked to have Celtic on easy street.
However, a lapse by Hoops defender Thomas Rogne five minutes later allowed Richie Foran to reduce the deficit before Grant Munro levelled from close range to keep Celtic two points behind their Old Firm rivals who have a game in hand.
The Parkhead side were booed off the park as Caley, who have now gone almost a whole calendar year unbeaten away from home in the league, celebrated in front of their small band of fans.
For the first time in a while this was all about football.
With Scottish grade one referees on strike in protest at increasing abuse and criticism, referee Alain Hamer and his team took charge and much of the attention was on the Luxembourg officials who took photographs of the stadium during their pre-match inspection.
Celtic, with a historical suspicion of Scottish officials, had led that criticism and thus the foreign officials received a big cheer from the home fans when their names were announced before the kick-off.
This week the jeers were reserved for their own side.
Amid the furore surrounding referees, though, it had been almost forgotten that a win for Celtic would take them above Rangers at the top of the SPL, at least until the Ibrox side's game against Dundee United at Tannadice the following day.
Shaun Maloney had a close-range shot smothered by Caley keeper Ryan Esson after only two minutes but the Hoops skipper injured himself in the process and had to be replaced by McCourt seven minutes later.
At the other end, moments later, Parkhead keeper Fraser Forster had to throw himself to his right to prevent Adam Rooney's curled effort from 12 yards going in at the far post.
Celtic began to turn the screw with Inverness forced to defend with some desperation at times although the Highlanders did look dangerous on the occasional counter.
Daryl Murphy was ruled offside when he had the ball in the net after 22 minutes but there was no great protest.
However, in the 37th minute the home side took a deserved lead through Ki.
McCourt broke in the box and when the ball fell to Ki, the South Korean quickly drilled a right-footed shot in off the near post.
The visitors started the second half in more positive fashion, pushing Celtic back towards their own penalty area.
Immediately a nervousness enveloped a less-than-full Celtic Park which led to Caley growing in confidence.
It had all fallen flat for the home side.
Youngster Richard Towell replaced Jos Hooiveld in the 56th minute as Lennon shuffled things around again and moments later Cha Du-Ri burst forward to send in a drive from distance which Esson almost let past him.
In the 65th minute Celtic looked to have sealed the points with a trademark McCourt solo goal.
The Irishman picked up possession of a loose ball 35 yards from goal and slalomed into the box past three Caley players before calmly slotting past Esson.
But five minutes later Rogne's mistake at the back allowed Foran to round Forster and drive in off the near post.
Then, in the 83rd minute Caley levelled when Jonny Hayes' corner from the right was helped on by Rooney and Munro knocked it in at the back post with the home defence standing in disarray.
The Hoops fans turned on their players and Hamer's final whistle brought a chorus of boos.