The 25-year-old had been without a club after leaving the Warriors back in April.
McGuigan is now looking forward to helping Exeter maintain their impressive start to the season in the Aviva Premiership.
"I played against the Chiefs a couple of times last season with Glasgow and I know they are a great team, who play some really good rugby. When I got the message to say about coming here, I was just so excited," he told the club's official website.
"I've only been here a short time, but already I feel so welcome. All the boys are very friendly and I know one or two of the guys already, especially Moray Low who I played with at Glasgow. I'm looking forward to getting stuck into things."
Exeter are back in action on Saturday when they host Bath in the LV= Cup.