Simon Shaw was the unlikely catalyst for a Wasps attack from their own 22.
Monday night's match with Quins also saw the first appearance in a Wasps shirt of rugby league convert Lee Smith.
But a Wasps penalty try set up a tense finish, with Bristol replacement Ed Barnes landing an equalising penalty.
But, playing with very little possession in the second half, Quins had to dig in desperately to deny a Wasps side who had won seven of their last eight matches.
Rees' roll call of young stars - all of whom are products of the club's hugely successful Academy - is a reminder that Wasps have a huge pool of exciting talent to draw on.
Scrum-half Eoin Reddan and hooker Raphael Ibanez crossed for first-half tries and Leicester were unable to find a way back against a resolute Wasps outfit who led 13-9 at the break.
Among four all-Pemiership One ties, holders Wasps will meet Newcastle in a repeat of last year's final, while reigning league champions Leicester, who like Wasps received a bye into the fifth round, take on London Irish at The Stoop.
But Munster were unable to generate a scoring position as Wasps held on for a vital victory in a devilishly difficult group, the Irish marauders having to settle for a bonus point.
His father, a formidable shot and a hater of wasps, once triumphantly cut one in half as it flew, with a carving knife.
Dowd won 59 caps for the All Blacks and as a coach with Wasps, helped the club to the then Powergen Cup last year and a second Heineken Cup victory this May.
David Lemi's converted try then gave Wasps a 17-10 lead at the break.
Tom Varndell then went over to give Wasps a five-point lead at half-time.
The visitors made four substitutions nine minutes into the second half - and the changes gave Wasps a new impetus.
Ugo Monye and Gonzalo Camacho scored tries while Varndell crossed twice to give Wasps a 16-15 lead at the break.
Strettle would not have appreciated the bumper crop of supporters looking on as his mistake gave Wasps a valuable early advantage.
And the only second-half score was a penalty from Wasps' Dave Walder.
In the short term, he will miss Saturday's final against Gloucester at Twickenham, and the Heineken Cup final against Wasps a week on Sunday.
Tuitupou, who had over-run the original kick, pounced on the loose ball ahead of a desperately retreating Wasps defence to dive over the line.
Francisco Leonelli scored twice for the hosts but Raphael Ibanez, Tom Voyce, Dominic Waldouck (2) and Paul Sackey tries gave Wasps a 35-12 interval lead.
King's second penalty gave Wasps a glimmer of hope on 65 minutes, but Gloucester seemed to have killed off the game by claiming their third try.
James took the lead back out to six points with a penalty but Wasps never know when they are beaten in Europe and they went in front when Haskell drilled his way over for a converted try after Clermont lost control of a line-out.
The match turned nasty on the hour mark when Canadian flanker Jamie Cudmore unleashed a flurry of punches on Wasps prop Tim Payne, sparking a brawl.
Robinson missed a late penalty attempt but Wasps secured their first win in a month as Bath ran out of time to have a chance of completing a fight-back.
Ben Foden scored a fine try to open the Saints' account after 55 minutes but Cipriani nudged Wasps into a 9-5 lead and they held on to claim victory.
After Serge Betsen had gone close for Wasps, Northampton assumed control and several times went close to scoring, but a combination of solid Wasps defence and Saints errors kept them scoreless.
Wasps centre Steve Kefu had just fired a kick directly into touch when Robson became embroiled in an off-the-ball scuffle and aimed a head-butt at Wasps scrum-half Joe Simpson.
Wasps flanker Tom Rees could be in for a surprise recall if his Wasps team-mate Worsley fails to recover in time.
But their lead was short-lived as Davis slotted over a second penalty to ensure Wasps went in at half-time with a slender advantage.
Jon Clarke was parachuted straight into Northampton's side after a lengthy absence, and Wasps exploited his lack of match practice with a slick try on 12 minutes.
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