Celtic had taken an 18th-minute lead thanks to a curling Virgil van Dijk free-kick, before it was controversially denied a penalty on the stroke of halftime when Josh Meekings stopped a Leigh Griffiths header with his arm.
Greg Tansey then levelled from the spot in the 58th minute after Celtic 'keeper Craig Gordon was sent off for hauling down Marley Watkins.
Edward Ofere fired Inverness in front in the 96th minute before a blunder from Inverness 'keeper Ryan Esson allowed substitute John Guidetti's free-kick to bounce over him as Celtic grabbed a 103rd-minute equaliser.
With time running out, Raven blasted home a winner in the 117th minute as Inverness, who famously knocked Celtic out of the same competition in 2000 while a part-timer, claimed a place in the final for the first time in its history.
Inverness will face Falkirk on 30 May 30 athe Bairns defeated fellow-Championship side Hibernian 1-0 in Saturday's semi-final thanks to Craig Sibbald's headed goal.
