However, in the 26th minute, Anthony Stokes found room inside the Hoops' box to flick a header from Derek Riordan's corner past Artur Boruc.
Celtic regained the initiative after the break but in the first minute of two added minutes, Galbraith, on for Colin Nish after the interval, fired in a dramatic breakaway winner after being set up by John Rankin.
There was a hint of offside about the goal but the Hoops fans were just as angry at finding themselves 10 points behind Rangers albeit with a game in hand.
Celtic boss Tony Mowbray made one change to the side which beat St Johnstone 4-1 at McDiarmid Park on Sunday, left-back Paul Caddis coming in for stopper Glenn Loovens who sustained a face injury in Perth, with Darren O'Dea moving into the centre of defence to partner 18-year-old Josh Thompson.
With only one regular defender in Celtic's back four, right-back Andreas Hinkel, there appeared the opportunity for the Leith side to take advantage.
But before Hibs had a chance to test that theory, they found themselves behind.
In the fifth minute, after Celtic midfielder Aiden McGeady won a corner on the left, Fortune headed the home side into the lead.
When Caddis' whipped-in cross came over, the Hoops striker, who notched a double at St Johnstone, reacted first at the front post to head past Hibs goalkeeper Graeme Smith and in off the crossbar.
The visitors went to pieces for the next few minutes as Celtic threatened to increase their lead, Fortune going close with another couple of headed efforts.
The first time the visitors really threatened was midway through the first half when Stokes tried his luck from 30 yards but the ball sped straight to Boruc.
The game, though, was to take a sudden and surprising U-turn.
In the 26th minute, after Boruc had pulled off a fine save to prevent Thompson scoring an own goal with a header, Hibs drew level from the resultant corner.
Riordan crossed from the left and Stoke nipped in ahead of Boruc to flick his header into the net.
With Celtic rattled and the home fans growling in frustration at how the game had turned, the former Sunderland player drove just wide from distance.
Celtic regained the upper hand but Hibs had sorted out their defence.
In the final minute of the half, though, after Hibs skipper Chris Hogg had fouled Caddis 30 yards out, the ball ended up in the net from Marc Crosas' free-kick but referee Iain Brines ruled it out for an infringement.
The first half ended with Celtic striker Georgios Samaras picking up a booking for a late challenge on Smith - a decision which prompted the Hoops fans to jeer the official as the teams trooped off.
Within a minute of the second half starting, McGeady ran onto a Fortune pass but screwed his shot just wide of the target.
Then, as the home side increased the pressure, Samaras missed his kick altogether in front of goal from Niall McGinn's volley back into the danger area before Smith made a good save from McGeady's curling effort from the edge of the box.
Korean midfielder Ki Sung-Yueng replaced McGinn on the hour mark, seconds after the Northern Ireland international had a close-range volley brilliantly blocked by Smith.
Five minutes later Lewis Stevenson and Galbraith came on for Riordan and Nish, as the visitors tried to get a foothold in the game again.
But it was Samaras who missed a sitter when he headed Caddis' cross high over the bar from just six yards out, the Greece international rewarded for his mistake by being immediately replaced by Morten Rasmussen who made his debut.
Paddy McCourt replaced Zheng Zhi and soon after, as Celtic pounded the Hibs box, his corner was headed over by O'Dea at the near post.
In the 82nd minute, as Hibs broke, midfielder Liam Miller appeared to be pushed to the ground by O'Dea inside the Celtic box but referee Brines ignored penalty claims.
As tension grew inside Celtic Park, Hogg sliced his shot wide of the target from Galbraith's corner.
But in a sensational finale, with seconds remaining, Hibs broke and Galbraith slotted past Boruc for the winner.