A last-gasp try from Trevor Leota gave Wasps a thrilling victory to set up a Heineken Cup final against Toulouse.
HEINEKEN CUP SEMI-FINAL
Munster 32 (15)
Tries: Foley, Williams
Cons: Holland (2)
Pens: O'Gara (3), Holland (3)
Wasps 37 (17)
Tries: Lewsey, Volley, Van Gisbergen, Voyce, Leota
Cons: King (3)
Pens: King (2)
Munster trailed 15-17 at half-time but led by 10 points with 15 minutes to go after Anthony Foley and captain Jim Williams had both gone over for tries.
But with seconds ticking away Alex King converted Tom Voyce's try and six minutes into injury-time Leota scored.
It took the video referee to confirm the try and Wasps held on to reach their first Heineken Cup final.
Ronan O'Gara put Munster on the board with a penalty after just two minutes when Leota was caught offside.
But the visitors were soon ahead as the ball was spread down the back line, Stuart Abbott fed Josh Lewsey and the winger forced his way over. King converted the score.
The influential O'Gara struck twice with the boot either side of a penalty from King, but the Munster fly-half limped off after 33 minutes.
Jason Holland took over kicking duties with aplomb, giving the home side a 12-10 lead.
But Holland then saw a kick charged down by Rob Howley, who fed Paul Volley for a try - converted by King - on the stroke of half-time.
Despite having Joe Worsley in the sin bin, Wasps extended their lead two minutes after the break when Howley sent Mark Van Gisbergen over in the corner.
It was the cue for Munster to launch a stirring fightback as Holland cut the gap with another penalty.
And, with Wasps' Fraser Waters in the sin-bin, Stephen Keogh's barnstorming run set up a try for Foley.
The flanker appeared to spill the ball forward in the tackle, but the score was awarded as Foley followed up to touch down and Holland converted.
Four minutes later Munster had a 10-point lead as Donnacha O'Callaghan stole the ball at a line-out and Williams touched down.
But Wasps refused to buckle.
King kept his nerve to slot a penalty and, after Munster had O'Callaghan and then Rob Henderson sent to the sin bin, the dramatic comeback unfolded.
Voyce ran through to touch down between the posts, King's conversion levelled the scores, and then Leota just managed to get the ball down in the corner to seal a superb win.
Wasps: M Van Gisbergen; J Lewsey, F Waters, S Abbott, T Voyce; A King, R Howley; C Dowd, T Leota, W Green, S Shaw, R Birkett, J Worsley, P Volley, L
Replacements: T Payne, B Gotting, M Purdy, M Lock, P Richards, M Denney, A Erinle.
Munster: C Cullen; J Kelly, M Mullins, R Henderson, S Payne; R O'Gara, P Stringer; M Horan, F Sheahan, J Hayes, D O'Callaghan, P O'Connell, J Williams (capt), S Keogh, A Foley.
Replacements: J Flannery, G McIlwham, D Pusey, E Havley/A Quinlan, E Reddan, J Holland, A Horgan.
Referee: Nigel Williams (WRU).