The Premiership side, who are struggling in their domestic competition, needed only a victory to win their pool but managed to get a bonus point, scoring four tries, to secure a vital home draw in the quarter-finals.
Elliott Daly, Matt Everard, Tom Lindsay and a penalty try proved to be enough for the former Amlin Challenge Cup Champions, who lifted this trophy in 2003.
The win keeps Wasps at the pool summit despite Bayonne's thumping 92-6 win over minnows Rovigo.
The French side, who are 9th in the Top 14 in France, made the perfect start against when they opened the scoring with an interception try after only six minutes.
Wing Vunga Lilo took full advantage of the visitors' aggressive defence to score the try to stun the home crowd before centre Daly showed impressive pace and footwork to run in from 30 metres for Wasps' first try.
Fly-half Ryan Davis, who had missed an early penalty and his conversion attempt, finally found his kicking boots to slot over a penalty and give the home side an 8-7 lead midway through the first half.
Wasps created plenty of chances and enjoyed dominance at scrum and lineout but lacked the real cutting edge against a Bordeaux defence which was stubborn and resilient for most of the second half.
Flanker Everard crashed over from short range from a 10-metre scrum after some good work by scrum-half Joe Simpson, who combined well with full-back Jack Wallace to give Wasps a 13-7 lead at half-time.
Fly-half Camille Lopez slotted over a penalty to claw Bordeaux-Begles back to within three points just after the break before Davis replied for London Wasps with his second penalty.
Full back Laurent Ferreres then landed a mammoth 55-metre penalty to haul the French side back into the contest before Davis landed two penalties to make it 22-13 midway through the second half.
Wasps were then awarded a penalty try as visiting number eight Hugh Chalmers was yellow-carded in the 72nd minute after repeated infringements by his pack. Davis added the conversion, making it 27-13.
Prop Lindsay then scored the vital bonus-point try in the dying moments to seal the victory and a home quarter-final in European rugby's second-tier tournament.
The French club knew they would have to win and rely on Bordeaux-Begles causing an upset against Wasps to have a chance of advancing.
They completed their part of the job and cruised to a 14-try victory over the Italian side but it was not enough as Wasps won 36-13 to top Pool Three.
Bayonne raced into a 33-6 half-time lead with tries from David Roumieu, Guillaume Bernard, Mark Chisholm, Joe Rokocoko and Thibault Lacroix.
The floodgates opened in the second half as the Top 14 outfit ran in a further nine tries including a hat-trick for Francois Carillo.
Clement Lagain, Otazo, Rokocoko, Mike Phillips, Marvin O'Connor and Julien Audy also crossed, with Otazo converting 11.
Rovigo's points came from two first-half German Bosto penalties.Wasps.