Flanker James Haskell and full-back Danny Cipriani grabbed the final touchdown for the depleted side, with two tries by their Irish scrum-half Eoin Reddan and 15 points from the boot of fly-half Alex King ensuring victory.
Wasps will be away for the winners of tomorrow's quarter-final between French side Biarritz and Guinness Premiership strugglers Northampton.
But it was never as easy as the scoreline suggests for the 2004 Heineken Cup winners who had to battle to break down a Leinster side lacking their inspirational Ireland skipper Brian O'Driscoll.
Wasps opened up a 6-0 lead with two 40-metre penalties by King inside the first 15 minutes. Then King's tackling played a big part in stifling the dangerous Leinster backs.
He took out their Argentinean fly-half Felipe Contepomi and Ireland winger Shane Horgan with pinpoint challenges as Leinster had their attacking attempts disrupted.
King was just short with a third penalty attempt, from 45 metres, and Contepomi missed a much easier kick before Leinster stormed back into the game with a try which came out of virtually nothing.
Opportunist scrum-half Chris Whitaker spotted a slim gap and made the most of the close proximity of referee Nigel Owens as he raced through unchallenged in the 23rd minute to leave Contepomi with an easy conversion.
But Wasps regained the lead in the 39th minute when a pass from Leinster number eight Jamie Heaslip went to ground in his own half and Reddan snatched up the ball to outpace the visiting defence on a 65-metre dash for the line.
King converted to put the English side 13-07 ahead but Wasps had to endure a tremendous attacking assault during the 11 minutes of first half injury time.
Whitaker almost found his way through again to set up a wave of attacks which saw Wasps in on their own line, defending desperately to subdue the forays led by heavyweight Samoan prop Stanley Wright.
The defensive effort cost them the services of former England skipper Dallaglio, yellow-carded for preventing Wright from recycling the ball, with Contepomi kicking the resulting penalty.
But the temporary loss of Dallaglio failed to take the sting out of Wasps who scored two tries while he was off the field.
The inspired substitution of Haskell for fellow England back row player Joe Worsley proved a match-winner.
Haskell, who will be 22 on Monday, celebrated his birthday early, taking a pass from lock Tom Palmer to touch down within two minutes of taking the field.
Ten minutes later Wasps put the pressure on again with Haskell turning provider to release 19-year-old Cipriani, preferred to England international Mark van Gisbergen in for Wasps third try.
Contepomi pulled back a 63rd minute penalty for Leinster but, two minutes later, King landed his first penalty of the match to restore Wasps' 15-point lead after Malcolm O'Kelly was shown a yellow card.
He joined Wasps centre Dominic Waldouck in the sin-bin after he had impeded Heaslip as he chased a kick five minutes earlier.
Reddan finally took the match beyond the reach of his countrymen with his second try in the 69th minute, with Haskell once again providing the short final pass to release the scrum-half on his 25-metre run to the Leinster line.
The Irish side never gave up battling but the Wasps defence held firm to make sure they followed Munster, beaten by Llanelli last night, out of the Heineken Cup to end Irish hopes for this season.
Wasps: Danny Cipriani; Josh Lewsey, Fraser Waters, Dominic Waldouck, Tom Voyce; Alex King, Eoin Reddan; Tim Payne, Raphael Ibanez, Phil Vickery, Simon Shaw, Tom Palmer, Joe Worsley, Tom Rees, Lawrence Dallaglio (c)
Replacements: Joe Ward, Peter Bracken, Dan Leo, James Haskell, Mark McMillan, Ayoola Erinle, Mark Van Gisbergen.
Leinster: Girvan Dempsey, Shane Horgan, Kieran Lewis, Gordon D'Arcy, Denis Hickie, Felipe Contepomi (capt), Chris Whitaker, Stanley Wright, Bernard Jackman, Will Green, Trevor Hogan, Malcolm O'Kelly, Stephen Keogh, Keith Gleeson, Jamie Heaslip.
Replacements: Brian Blaney, Reggie Corrigan, Owen Finegan, Cameron Jowitt, Guy Easterby, Christian Warner, Rob Kearney.