First-half tries from Jack Yeandle and Jack Nowell helped the visitors take a 20-3 lead, but a strong finish to the opening period led to a Christian Wade try for Wasps, leaving them just 10 points down at the break.
Damien Welch added Exeter's third shortly after the restart, but James Scaysbrook was sent to the sin bin shortly after, allowing Hugo Southwell to cross the whitewash and keep the home side in the match.
However, Haydn Thomas managed to restore Chiefs 17-point lead before Scaysbrook returned to the pitch, while some solid defending by the visitors ensured that the hosts could not get back in the match.
Tom Varndell scored a consolation try in the 79th minute, with Exeter having taken off many of their first-choice players.
The result moves Exeter up to sixth in the table, while Wasps remain eighth.