But as the 14 men tired, Wasps cranked up the pressure in the closing minutes and won a penalty try from a driving line-out maul which denied them even a losing bonus point.
The game's opening try came on 15 minutes, veteran prop Budgen emerging with the ball after Exeter had produced a superb driving maul from a line-out on 18 metres.
They seemed to be feeling the effects of the quick turnaround and Tonga made an explosive start with Toulouse's Maka going over after a rolling maul from a line-out after 72 seconds.