Alexander Radulov scored two goals for Dallas, and Anton Khudobin made 40 saves.
Vladislav Namestnikov scored twice, and Nazem Kadri and Andre Burakovsky each had a goal and an assist for Colorado, the No. 2 seed. Michael Hutchinson, the third-string goalie, made 30 saves in his third straight start.
Colorado was without goalies Philipp Grubauer and Pavel Francouz, forwards Gabriel Landeskog, Joonas Donskoi and Matt Calvert, and defenseman Erik Johnson. Each was unfit to play.
"I know it's not the Cup Final or anything, but we felt like we could win and I still feel like we could have won. It's just unfortunate," Avalanche forward Nathan MacKinnon said. "I would have liked to see our team fully healthy, that's for sure. Lost a lot of key guys. That's the way it is. You can call it an excuse or whatever you want, but that's just the way it is. We lost a lot of key guys to our team."
Kiviranta tied the game 4-4 with 3:30 left in the third period, 10 seconds after Namestnikov scored with 3:40 remaining for a 4-3 lead.
Kadri made it 3-2 at 5:45 of the second period on a power play, his ninth goal of the playoffs, before Radulov tied it 3-3 on a power play at 11:28 of the third, his fifth goal of the series.
Radulov scored a power-play goal off a loose puck in front of the net at 2:39 of the first period for a 1-0 lead.
"It's a Game 7. It doesn't matter who goes out there," Radulov said. "I'll take it, but it doesn't matter who's scoring. What matters is we move on and win the series. That's the big thing for us."