The 29-year-old has clocked up eight seasons with Warriors after signing on back in 2006.
He has gone on to make 144 appearances for the RaboDirect Pro12 side as well as earning 25 caps for Scotland.
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 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.”
Exeter coach Rob Baxter, who will be losing tightheads Hoani Tui and Craig Mitchell at the end of the season, said: “It is obviously a very good signing for us as we are losing a couple of quality tightheads so replacement has been a big part of our recruitment.
“We want to be a very good Premiership club so we have to recruit players that we feel fit in with what we want to do here but who will also add things.
“I think he is a good quality signing and a guy who has not yet peaked as a tighthead prop.”