Exeter booked an Aviva Premiership final appointment with Saracens at Twickenham next Saturday after continuing their remarkable rise towards the summit of English rugby.
The Chiefs only secured Premiership status in 2010, but they are now 80 minutes away from possibly being crowned league champions as Wasps were sunk 34-23 thanks to two penalty tries and touchdowns by centre Ian Whitten and flanker Dave Ewers.
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Fly-half Gareth Steenson kicked four conversions and two penalties, while his opposite number Jimmy Gopperth booted 13 points, including conversions of tries from hooker Carlo Festuccia and scrum-half Dan Robson.
And it means that Exeter now stand between Saracens and them achieving the first domestic and European double by an English club since Wasps 12 years ago.
Saracens will start as red-hot favourites in defence of their Premiership crown - they beat Exeter 36-18 at Allianz Park eight weeks ago - yet the Chiefs are unlikely to show any respect for reputations.
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Wasps pushed them to the limit, but Exeter showed enough collective composure when it mattered in stamina-sapping conditions as they sent a capacity Sandy Park crowd of just under 13,000 wild.
Exeter showed a number of changes from the side that crushed Harlequins in their final regular season league game two weeks ago, with Whitten, hooker Luke Cowan-Dickie and lock Geoff Parling among those called up, while Wales prop Tomas Francis featured on the bench after completing an eight-week ban imposed during this season's Six Nations.
Wasps, meanwhile, made one switch following their European Champions Cup semi-final defeat against Saracens in Reading last month as prop Jake Cooper-Woolley returned from injury to pack down alongside front-row colleagues Festuccia and Matt Mullan.
Exeter made a bright start amid torrential rain and a gusting wind, moving through the phases accurately and patiently before Wasps conceded a fourth-minute penalty when their number eight Nathan Hughes infringed.
Steenson, one of only two survivors in Exeter's starting line-up from the team that clinched Premiership promotion by beating Bristol six years ago, stepped up to accept the chance, and Exeter were off and running.
Steenson doubled Exeter's advantage eight minutes later, this time taking advantage of a Wasps scrummaging offence as the home side underlined early territorial control and gave their opponents plenty to think about.
But Wasps' response proved an impressive one, as their pack drove relentlessly from an attacking lineout before Festuccia got his hands on the ball and claimed a try that referee Greg Garner awarded following consultation with television match official Stuart Terheege.
Gopperth's conversion against the wind put Wasps a point ahead, yet Exeter almost replied immediately when a slick handling move saw Chiefs' England wing Jack Nowell freed in space, but Wasps full-back Charles Piutau completed a superb cover tackle.
It proved a brief reprieve for Wasps, though, with Nowell then popping up in midfield and finding Whitten in support, and the centre displayed impressive power to crash over from 15 metres before Steenson's conversion made it 13-7.
Yet Wasps had no intention of playing second fiddle, and they scored a second try eight minutes before half-time when strong approach work by George Smith, Hughes and wing Frank Halai ended with Robson darting over from close range, and Gopperth's conversion edged the visitors back in front.
The game continued to thrive at relentless pace, despite such testing conditions, and Exeter number eight Don Armand crossed Wasps' line, only for Garner to rule out the try after further TMO discussion, as a knock-on by Nowell was rightly identified during the build-up.
But Exeter did not have to wait much longer to claim an interval lead, with the Chiefs driving a close-range lineout and Garner punishing Festuccia - who was yellow-carded for pulling down the maul - and awarding a penalty try that Steenson converted for a 20-14 half-time advantage.
It appeared the correct call by Garner, although he missed an earlier incident when Exeter scrum-half Will Chudley's boot looked to make contact with Wasps and England lock Joe Launchbury's head, which could easily attract citing commissioner attention.
Gopperth opened the second-half scoring with a penalty, but Exeter hit back within two minutes with a show of brute strength as Ewers, who could be named in England's squad on Sunday to tour Australia, muscled his way over, with Steenson's conversion opening up a 10-point gap.
Another Gopperth penalty made it 27-20, before he struck again 13 minutes from time, ensuring that a tense battle of wits and nerve would be played out to the end.
But Exeter stayed calm, and a second penalty try - this time following a Wasps scrummaging offence - put the Chiefs in dreamland.