Moray joined the Warriors in 2006 and has to date made 144 appearances for the RaboDirect PRO12 side. He has gone on to win 25 caps for Scotland.
Moray Low said: "I've really enjoyed my time at the Warriors and I'd like to thank everyone at the club for helping me develop as a player.
"I'm excited about this new challenge, but we still have some very important games, as we look to try and reach the play-offs once again."
Glasgow Warriors head coach Gregor Townsend said: "We'd like to thank Moray for everything he's done for the club over the past eight seasons and we wish him well for his next challenge at Exeter.
"He has always been committed to helping the team and is a popular figure within the squad.
"I know he will be determined to end the season and his time at the club on a high and we look forward to seeing him back in a Warriors jersey when he returns from international duty.
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