Rivet took a shot off a foot during a power play midway through the third period and struggled to get back to the bench, while Dandenault came away from a corner hit limping later in the same period.
Montreal was forced to play the final few minutes of regulation and overtime with four defencemen.
"It wasn't easy, but we've all played lots of hockey before, so we were able to get through it," said Janne Niinimaa, making his home debut with the Canadiens.
"We had tied the game already, so we just wanted to make it to overtime and secure the point first, then try and get the second one."
The Canadiens are already thin on the blue line as Francis Bouillon continues his recovery from off-season knee surgery, so losing Rivet and Dandenault would be a tough blow.
Mark Streit is on the roster if either injury is serious enough to put Tuesday's home game against the Calgary Flames in jeopardy, but if both players are unavailable a call-up from the Hamilton Bulldogs would be necessary.
However, Canadiens head coach Guy Carbonneau did not appear too concerned when asked about the status of his defencemen.
"Lower body injuries," he said with a smile. "We'll have to see (Sunday) how they're feeling."
Canadiens captain Saku Koivu suggested the injuries would not prevent either Rivet or Dandenault from suiting up Tuesday night.
"You never want to see guys getting hurt," Koivu said. "But I don't see it being a problem for the next game."
Otherwise, top free agent acquisition Sergei Samsonov won over a sceptical public in Montreal with his game-tying goal at 16:37 of the third after struggling throughout the pre-season.
"It's nice to get it out of the way," Samsonov said. "We battled back and got a point out of it. I think we skated a lot better in the third."