Some clever set-piece play and effective kicking from Andy Goode helped the hosts to secure their second consecutive victory in the competition.
A couple of Goode penalties brought the score to within one, before Brett replied to extend the French side's lead at the break.
Shortly into the second half Wasps took the lead. After an early penalty, a well-organised lineout allowed Ed Jackson to go over before Goode dispatched the conversion.
Wade then released Bassett once more and this time he made it count sliding over the line and allowing Goode to add two more points to the score.
More good work from the front row then earned Wasps another penalty and Goode continued his kicking form, making it 26-10.
The home side piled on the pressure in the final few minutes with Bayonne down to 13 men but couldn't quite manage to add to their lead.