Jimmy Gopperth held his composure to nail a match-winning conversion with the final kick of an epic European Champions Cup quarter-final to break Exeter's hearts at the Ricoh Arena.
Gopperth added the touchline extras to full-back Charles Piutau's second try of the match - in the 79th minute - as Wasps edged through 25-24 before a crowd of 23,866.
Top try scorer in both the Premiership and Champions Cup Thomas Waldrom looked to have stung Wasps for a second time this season as Exeter built a commanding lead midway through the second-half.
Waldrom, who scored a hat-trick of tries when Exeter beat Wasps in an Aviva Premiership encounter four months ago, struck with two more touchdowns, while substitute prop Harry Williams also scored, with skipper Gareth Steenson kicking all three conversions before adding a penalty.
But Wasps were not to be denied, as Pituau struck in the dying seconds following a relentless spell of Wasps pressure that Exeter looked to have repelled through some sustained, heroic defending.
It followed the full-back's earlier score, while wing Frank Halai also touched down, with Gopperth adding two penalties and two conversions to book a semi-final appointment with either Saracens or Northampton at Reading FC and London Irish's Madejski Stadium on April 23.
Wasps made the initial headway in terms of territory, using their number eight Nathan Hughes to impressive effect, and they deservedly took a sixth-minute lead when Gopperth bisected Exeter's posts from 45 metres.
Exeter had barely been in the game until that point, and apart from a sharp break by their England wing Jack Nowell, the Chiefs continued to play a back-foot role as Wasps performed with greater precision and purpose.
The home side almost extended their lead when Tongan centre Siale Piutau burst through a sizeable gap in Exeter's defence, and the Chiefs only escaped after a Wasps knock-on close to their line, but Exeter then infringed again and Gopperth doubled Wasps' advantage through a second penalty.
Exeter's first notable excursion inside the Wasps 22 came following a Gopperth blunder when he he let a kick from his opposite number Steenson bounce, before it rolled into touch.
Wasps managed to clear the danger, with both sides also forced into early injury changes as Exeter's Scotland international prop Moray Low departed to be replaced by Williams, and Wasps lock Kearnan Myall was helped from the field, meaning an early appearance for Wales second-row forward Bradley Davies.
Waldrom made a couple of trademark surges that took the Chiefs close, yet Wasps' superior technical discipline kept them in charge before Exeter struck with two converted tries either side of a long-range Gopperth penalty hitting the post.
Waldrom did not require a second invitiation to power over when Exeter's forwards drove a lineout, and Steenson's conversion edged the visitors in front before Waldrom accepted a gift from Davies just 15 minutes after he arrived on the pitch.
Davies appear to have no idea what was going on around him, and a serious breakdown in communication was punished when the Welsh lock's loose pass was picked off by Waldrom who stormed 30 metres to the line, with his momentum taking him over despite the efforts of three Wasps defenders.
Steenson again kicked the conversion, and Exeter trooped off 14-6 ahead following a dominant period that left Wasps stunned.
But Wasps needed just three minutes of the second-half to regroup, and they did it in style through a brilliant try started by Siale Piutau's break, before scrum-half Dan Robson kicked superbly into space and Charles Piutau applied a muscular finish despite Nowell's attempt to haul him down.
Gopperth sent the touchline conversion attempt wide, but before Wasps could plot another raid on Exeter territory, the Chiefs replied with a third try as Williams gained the touchdown. Steenson slotted the easy conversion to open up a 10-point lead.
Exeter then thought they had scored again, Geoff Parling breaching Wasps' defence out wide, but Poite ruled the try out for a forward pass, although the Chiefs' attacking intent suggested it was only a matter of time before they struck again.
A Steenson penalty took Exeter out to a 24-11 lead, only for Wasps to turn the game on its head with a quality try following a prolonged spell of Chiefs pressure as Davies proved an impressive link-man after Elliot Daly's sprint from deep in his own try area, with Halai going over and Gopperth's conversion setting up an intense finale.
Exeter looked to have done enough, but Wasps kept battering away, and they finally got their reward as Piutau and Gopperth combined to secure glory and the first semi-final spot in this year's competition.