Returning Hoops midfielder Tom Rogic gave the visitors the lead in the 17th minute before loan forward Billy King levelled with a powerful drive 10 minutes later.
In the 33rd minute Scott Sinclair kept his goal-a-game record in the league going with a fine finish and only Inverness keeper Owain Fon Williams' fine display - and the woodwork - prevented the Hoops adding to their lead after the break.
Fisher, on for Lonsana Doumbouya, then dramatically headed the equaliser to take the Highlanders off the bottom of the table.
Brendan Rodgers' side go three points clear of Hearts at the top of the table with a game in hand and will look at this as a blip as they go for a sixth successive title.
Rogic was one of two changes made by Brendan Rodgers from defeat to Barcelona, with midfielder Callum McGregor also in as Nir Bitton dropped to the bench and 35-year-old defender Kolo Toure was rested altogether.
Right-back Cristian Gamboa made his domestic debut and long-term injury absentee Jozo Simunovic was named among the substitutes, looking to feature for the first time since January 23.
For the home side, French forward Doumbouya replaced Fisher who started on the bench.
Inverness looked confident and keen in the early stages but were found wanting in defence when Rogic had all the space he required 12 yards from goal to volley in a cross from Hoops left-back Kieran Tierney, who had been picked out by Erik Sviatchenko.
It was the Australia international's fifth goal of the season but it was nullified when King drifted inside Gamboa from the left and fired in a dipping drive from 25 yards which Dorus de Vries could only help into the net.
Moments later Caley Thistle had a clear-cut chance when midfielder Ross Draper dispossessed McGregor and raced clear.
As the midfielder entered the opposition penalty box he went crashing to the ground under a challenge from the backtracking Sviatchenko but referee Don Robertson ignored what appeared to be a good penalty claim.
Sinclair was much more ruthless when he took advantage of a Josh Meekings mistake minutes later, driving to the edge of the Inverness box before curling the ball into the corner to maintain his impressive strike rate.
Celtic dominated the start of the second period but Fon Williams dived to his left i n the 50th minute to parry a powerful drive from Moussa Dembele, before tipping a Tierney piledriver over the bar for a corner which came to nothing
McGregor then crashed a shot off the top of the bar with the help of a deflection before Fon Williams pushed a Rogic free-kick away from danger.
The visitors were relentless and Sinclair drew another fine save from Fon Williams with a long-range effort before Simunovic replaced Gamboa to complete his long rehabilitation
Sviatchenko headed a Mikael Lustig cross off the post and Fon Williams clawed a McGregor chip away from goal after good footwork from the midfielder.
However, there was to be a late and unexpected twist.
With moments remaining of the regulation 90 minutes, Fisher leapt inside the Celtic box to head in a Greg Tansey cross to stun the visitors.
Tansey then curled a free-kick just past the Celtic goal but the Hoops might have snatched the win back had Fon Williams not saved an injury-time header from Dembele.
Source : PA
Source: PA