The injury could also affect McBeth's participation in Scotland's autumn Test series which begins a week on Saturday against Fiji and continues with games against South Africa, Portugal and Australia.
"Unfortunately, he did his MCL on his knee so he'll need to nurse himself back to recovery," said Glasgow assistant coach Nigel Carolan. "We're fortunate that we've got Rory Sutherland and Jamie Bhatti with us here and they're both raring to go."
Carolan also reported that scrum-half George Horne had sustained a "bang on his leg" during the game in Durban.