The dramatic late score gave the 2004 and 2007 champions a win that broke the hearts of the first-time quarter-finalists.
Wasps will now play at the Madejski Stadium on April 23 against Saracens, who overcame a miserable first half to beat Northampton and win their fourth quarter-final in a row.
Exeter's Thomas "The Tank" Waldrom scored two triesinside five second-half minutes to give the visitors control against Wasps and take his competition tally to six.
However, twice champions Wasps started the second halfstrongly and reduced Exeter's advantage when Piutau crossed the line following Dan Robson's fine chip.
Exeter reasserted their control four minutes later through Harry Williams but Frank Halai took Wasps within touching distance of an unlikely comeback with a 67th-minute try before Piutau and Gopperth finished the job in a breathless finale.
"I'd have preferred it if it didn't take us until 79minutes," said Wasps director of rugby Dai Young. "It would have been nice if it had been 29 minutes or so."
Northampton enjoyed a 10-6 halftime lead over Saracens following Ken Pisi's 17th-minute try at Allianz Park but were made to rue a series of missed chances.
Premiership leaders Saracens started poorly but imposed themselves on the game after the break and took the lead in the 67th minute when Chris Ashton powered over the line and Owen Farrell converted.
Chris Wyles added another try as Saracens extended their100 percent record in the competition this season, before Courtney Lawes' late consolation for Northampton.
"I asked for a big performance, and they gave me that," Saints skipper Ben Foden told Sky Sports. "We gave it our best shot but we came up short."
(Editing by Ken Ferris)
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