The Coventry-based side have invested heavily during the close-season and the returning Danny Cipriani was among the starting XV at the Ricoh Arena.
But Thomas Waldrom, the Premiership's top try scorer last season, and Will Chudley scored tries while Henry Slade contributed seven points to leave 2015-16 runners-up Exeter 17-8 in front by the break.
Jimmy Gopperth's try had kept Wasps in it, though, and he added seven from the tee in the second half, while scores from Dan Robson and debutant Tommy Taylor completed the turnaround.