Marriot has made 56 appearances for City since joining in 2007, before becoming the club's general manager.
Exeter boss Paul Tisdale said: "Andy has been a great asset to us, both as general manager and in the first team."
He added: "It's no surprise to us that he's got a job at a Premier League side and I'm sure he will be a huge success there."
Marriott joined the Grecians from Boston United in 2006 and was a key member of the side that won promotion into League Two.
The veteran keeper started his career as a trainee with Arsenal before joining Nottingham Forest in 1989.
He went on to have spells at clubs including Wrexham, Barnsley and Torquay before joining Exeter, and won five international caps for Wales.
Source: BBC Sport
Source: BBC Sport