The openside damaged ligaments after he got caught in an awkward position in the ruck and immediately signalled for help. He was treated on the pitch for a lengthy period and received oxygen before being wheeled off on a stretcher.
He was taken to hospital and is likely to have surgery on Saturday.
“I’m devastated for Rory Darge,” said Glasgow coach Franco Smith. “He’s a top player. It was an ankle injury and he will likely be operated on in the morning.
“It was a dislocation of the ankle. It’s not a season-ending injury, but I don’t want to put an exact timeline on when he will be back.”
The national side open their four-match autumn series against Australia at Murrayfield on October 29, then host Fiji, New Zealand and Argentina.
Glasgow were highly impressive in their first home match since Smith took over as head coach. They scored eight tries as they bounced back from last week’s 33-11 defeat by Benetton in the United Rugby Championship season opener in Italy.
“Obviously I’m very happy for the players,” said Smith. “I think they worked hard and wanted to see a result.
"The fact they were still throwing the ball around in the 84th minute showed their eagerness to play. All the coaches should be complimented because they worked really hard this week to put out the best product.”