The 2-1 success came from nowhere as the match looked certain to end with a shock victory for the Highlanders, who have twice dumped the Hoops out of this competition in the last seven years.
Celtic, Scotland's champions-elect, have built a reputation over the years for their never-say-die attitude and this was a classic example.
To not even get a draw from the game clearly sickened the hosts and manager Charlie Christie admitted later that tears were shed in the changing room due to the manner of this defeat.
Celtic only last month clinched a 2-1 SPL win with a 91st-minute winner and this compounded the misery for Inverness.
Graham Bayne continued his fine scoring run against the Old Firm when his first-half strike put his side ahead, but late, late responses from Scotland pair Steven Pressley and Kenny Miller gave Gordon Strachan's men the most unlikely of results.
The first opportunity fell to Miller after just two minutes when he beat the offside trap to home in on goal, but a decent block by keeper Michael Fraser denied him.
The opening goal came on 18 minutes when wonderful work from Barry Wilson saw him turn Pressley inside out in the area and tee-up Bayne, who made no mistake from six yards out.
This was just what the game needed and Inverness soon settled and gave the SPL's top team problems.
Celtic's key threat appeared to by Shunsuke Nakamura and his tested Fraser with a drive from distance as the looked for a way back.
Bayne then thought he'd doubled the lead when he netted from close range after an Ian Black shot was spilled by Artur Boruc, but he was flagged for offside to cut short his celebrations.
Celtic started the second half on the front foot and Nakamura rattled the crossbar from distance with a sublime shot.
Opportunities were few and far between after this until the drama unfolded with 90 seconds of regulation time left.
Nakamura flighted in an inviting corner which was headed on by Stephen McManus and Pressley was on hand to bury the ball in the crowded penalty box.
Just as Inverness were coming to terms with this blow and the possibility of a Parkhead replay, Miller conjured up the sucker-punch when he fired a terrific swerving shot beyond Fraser for a place in the last four.