Wasps trailed 10-6 at half time after an error-ridden first 40 minutes but quickfire tries from Ed Jackson and Josh Bassett midway through the second period ensured a comfortable finish.
Andy Goode contributed 16 points with the boot through four penalties and two conversions as Wasps built on last week's record-breaking 90-17 win at Viadana.
There was further good news for the victors - England flanker James Haskell made his first appearance since knee surgery with a 20-minute stint as a replacement.
Dai Young's men began badly as their stuttering start to the Aviva Premiership threatened to show once more when Bayonne wing Sam Gerber raced over inside five minutes. Matthieu Ugalde's scoring pass appeared to be forward but referee Peter Fitzgibbon decided otherwise and ignored the chance to call for the TMO.
Stephen Brett's conversion made it 7-0 but Wasps would have been celebrating a try of their own with 10 minutes gone had wing Bassett not fluffed a gilt-edged chance following Christian Wade's first mazy run.
Kearnan Myall was held up over the line as Wasps again went close with 14 minutes played but Goode slotted two penalties with 15 and 24 minutes played to cut the gap to a single point at 7-6.
Bayonne dominated the rest of the half, though, with only a last-gasp tackle from Wade preventing Scott Spedding from scoring before giant number eight Opeti Fonua was held up twice over the line.
It was all Wasps after the break as Goode's third penalty made it 10-9 after 44 minutes before the game was turned on its head in the space of just three minutes.
Jackson powered over from a driven lineout to give Wasps the lead for the first time after 52 minutes and Bassett put the game to bed when England wing Christian Wade produced a moment of magic to create the space to set him free moments later.
Goode converted both scores and kicked a final penalty just past the hour, although a try-scoring bonus never looked likely.
And on Friday, six first-half tries sent Grenoble to a rousing 40-7 hammering of Viadana in the same group at Stade Lesdiguieres.
- Sports & Recreation
- Andy Goode
- Josh Bassett