Dolly, who was nurtured in the Mamelodi Sundowns academy, collided with Strasbourg midfielder Jeremy Grimm a few minutes into the game and he was substituted in the sixth minute.
His South Africa team-mate Lebogang Mothiba, who recently joined Strasbourg from Ligue 1 outfit OSC Lille, came off the bench to score a late equalizing goal in stoppage time to ensure that the game ended in a stalemate.
Dolly underwent a successful operation on his fractured fibula on Monday and he is set to be out for several weeks.
Bafana Bafana have back-to-back fixtures against Southern African rivals Seychelles in October and three-time Afcon winners Nigeria in November.
"Keagan Dolly had a successful surgery yesterday (Monday). We wish him a speedy recovery," Montpellier said on their official Twitter account
Dolly was looking forward to playing regularly for La Paillade, who are currently placed fifth on the Ligue 1 standings.
"I've had a good pre-season and would love to enjoy a good run with Montpellier this season," Dolly told BBC Sport.
Dolly has made three appearances in the Ligue 1 this season plus one in the French National Three for the club's reserve team.
The left-footed attacker joined Montpellier from Sundowns midway through the 2016/17 season for a fee of around $2 million.