The scrum-half initially signed a one-year pro deal with Glasgow for the 2017/2018 season.
Speaking to Warriors TV, George said: "I'm delighted to be staying at Scotstoun. Growing up I always wanted to play for Glasgow Warriors and to get a good chance this season has been a lot of fun and I can't wait for next year now.
"I was targeting the November and 6 Nations periods for game-time this season, but I've been fortunate that the coaches have rewarded performances in training and it's been great to get so many minutes under my belt.
"Mike Blair has really helped me this season and it's been really positive, but there are areas of my game I need to work on and hopefully I can continue to do that.
"I'm just taking it one game at a time. My main goal at the moment is to get as many minutes under my belt as possible and contribute well to the team."
Glasgow Warriors head coach, Dave Rennie said: "It's brilliant that George is staying with us next season. I think he has been excellent this season and we'd expect him to get better and better.
"We need real depth in all positions and we're pretty fortunate with the scrum-halves we have.
"George is a rare breed who can run all day and run at pace. He suits our game, he gets to the break-down really quickly and has an ability to get a second touch and anticipate breaks, which has resulted in a lot of tries for him.
"He still has a lot to work on and he's fortunate we've got some very good nines who work together to develop each others games, but he's certainly put a lot of pressure on the more experienced players and he's got a lot of starts because of it."