A goal from substitute Jackie McNamara and two from Henrik Larsson gave Martin O`Neill`s side a place in the history books at Parkhead.
McNamara came on for the talismanic Lubo Moravcik early in the game with just 18 minutes on the clock, after the Slovak suffered a recurrence of his knee injury.
However, the Scottish international proved to be a true super-sub by scoring the first and setting the second up. Celtic had rarely been troubled in the first period and took a deserved lead just six minutes before the interval.
Former Hibees star Didier Agathe did all the hard before sending McNamara (pictured) through on the left-hand side and the former Dunfermline ace showed great technique to beat Nick Colgan.
Alex McLeish will have been demoralised to see his side concede such a late goal in the half, when it looked they could hang-on until half time, although just three minutes into the second period the task was made twice as hard.
Larsson had been looking to increase his 51 goal tally for most of the opening period and he made no mistake when McNamara sent him into the area and he blasted past Colgan, after Canadian international Paul Fenwick slipped and gave him a clear run on goal.
The Hibs fans and more importantly the players were stunned and Celtic just passed the ball around comfortably looking for a third.
The Edinburgh side did push forward looking for a consolation, but it was the Glasgow outfit that completed the scoring, when fittingly Larsson wrapped up a brilliant season for himself and Celtic with a spot-kick just minutes from time.
Ex-Aberdeen stopper Gary Smith was adjudged to have felled Larsson and the Swede picked himself up to dispatch his 52nd goal of the season to confirm himself the 'goal king of Europe'.
The Bhoys chief was understandably delighted with his side's performance which sees him go down in Celtic folklore. "I thought they were brilliant from start to finish," beamed O'Neill.
"It's been a long season and I thought (Chris) Sutton, who hasn't played a competitive game for a while, was absolutely essential for us.
"But to mention one player is wrong because they were all absolutely immense."
