Anton Khudobin made 38 saves, and Alexander Radulov had a goal and an assist for Dallas, the No. 3 seed in the West.
The Stars have won five straight games dating to Game 4 of the first round against the Calgary Flames.
"We've had some comeback wins," forward Joe Pavelski said. "… There's no quit in our]*
Trailing 2-0 in the second period, Dallas scored four times in 9:20, including two power-play goals in 43 seconds to tie the game. Pavelski cut the lead to 2-1 at 9:54 when he scored with a one-timer during a 5-on-3 advantage, and Radek Faksa scored on a goalmouth scramble at 10:37 to tie it 2-2.
The Stars killed a 5-on-3 Colorado advantage of 57 seconds after Andrew Cogliano was called for slashing at 11:05 and Blake Comeau was penalized at 12:08 for delay of game for playing the puck over the glass.
"That was huge," coach Rick Bowness said. "We needed that badly. That carries over into the penalty kill. It got us back into the game, clearly. Now the penalty killers go out and they know they have to do their job, and they did that."
Radulov scored on a deflection in front for a 3-2 lead at 15:34. Jamie Benn's shot hit Avalanche defenseman Samuel Girard's stick before bouncing off Radulov's shoulder and past Francouz.
Francouz was making his first start since a 4-2 loss Aug. 15 in Game 2 of the first round against the Arizona Coyotes. Goalie Philipp Grubauer is out indefinitely with an injury he sustained in the second period of a 5-3 loss in Game 1 on Aug. 22.
"This is something I've always dreamed about, a chance for me," Francouz said. "It depends on me how I take this responsibility and help the team win four more games. I'm into it. I know I can help the team more. Just help the team to win, that's my goal."
Esa Lindell gave the Stars a 4-2 lead at 19:14 of the second.
"My goal is to make sure our team is responding after goals-against or a bad bounce here or there, and we still have this belief that we can bounce back from things that are thrown at us," Avalanche coach Jared Bednar said. "There are a lot of things I liked, it's certainly a step forward from Game 1, but now we've put ourselves in a hole."