Cusiter took a quick penalty and went to ground awkwardly after being tackled under the posts - being helped from the pitch and then taken off to hospital.
The injury could mean Cusiter is unavailable for Scotland's autumn internationals against New Zealand on November 11 and then South Africa and Tonga thereafter.
Glasgow went on to beat Ospreys 28-10 and Glasgow coach Gregor Townsend admits the prognosis may not be good for both club and country.
"It is a worrying situation," said Townsend. "Chris went straight to hospital and we are all waiting to hear the extent of the damage."