The winning goal came from David Raven with three minutes remaining in extra-time, the full back slamming the ball home from a tight angle for his first strike of the season.
The win for the Highlanders denied Celtic the chance of claiming the treble, with Ronny Deila's men leading the Scottish Premiership having already lifted the League Cup.
The sides were level at 2-2 after 90 minutes, Virgil van Dijk opening the scoring for Celtic in the first half.
Greg Tansey equalised from the penalty spot just before the hour after Celtic goalkeeper Craig Gordon was sent off.
Edward Ofere put Inverness into an extra-time lead before John Guidetti responded with a 20 metre free kick.
"It's a dream. We deserve to be in the Cup final," Inverness manager John Hughes told the BBC.
Inverness will play Scottish Championship (second tier) Falkirk in the final at Hampden Park on May 30.
(Reporting by Douglas Beattie; Editing by Michael Hann)
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