Low, who has been a Warrior for eight years, said: "I've really enjoyed my time with Glasgow 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 Pro12 play-offs again."
Rob Baxter, the Chiefs director, believes that Low's experience can bolster the Sandy Park side's bid to challenge for Aviva Premiership honours.
Baxter said: "Moray is one of those guys who has not played masses and masses of rugby, but he has already got some very good experience at Heineken Cup and international level. Scrums are a massively important thing, and Moray is one of those guys who will really develop within the Premiership."
Low will find out tomorrow if he has retained his place in the Scotland line-up for Saturday's RBS 6 Nations match with Italy in Rome.