Number eight Alfie Barbeary's late effort sent Wasps through to the European Challenge Cup semifinals with a 34-30 win at Edinburgh on Saturday.
The 2003 champions will play Lyon or Glasgow in the last four next weekend with those outfits meeting later in the day.
In the Scottish capital, after an open first half the hosts led 17-10 thanks to scores for Ben Vellacott and Adam McBurney as Jimmy Gopperth responded for the English outfit.
With less than 10 minutes to play the sides were level 27-27 before Argentina full-back Emiliano Boffelli kicked a penalty to give Edinburgh a three-point lead.
Former England under-20's hooker Barbeary's key intervention came with three minutes remaining as he crashed over from a short range pick and go to claim Wasps' first European semi since 2016.
On Friday, Saracens hammered fellow English club Gloucester 44-15 to book their place in the next round.
The three-time Champions Cup winners will face either Toulon or London Irish who meet in the fourth quarterfinal on Sunday in the south of France.