Kadri redirected a wrist shot by Kevin Connauton at 13:54 for a 5-4 lead after Mikko Rantanen scored for Colorado to tie the game 4-4 at 12:02.
Dallas scored three straight goals in a span of 6:05 in the third period to take a 4-3 lead.
It was the first time in the history of the Stanley Cup Playoffs that a team had a two-goal lead in the third period, fell behind, and then restored its two-goal lead.
"I think it's a character win," Kadri said. "It would have been easy for us to get down on ourselves, but we understand we can score goals … we're pretty comfortable playing in every situation. Obviously blowing the lead, it wasn't our best, but this is a game that can definitely give us some momentum and hopefully change a series."
Kadri, Rantanen and Nikita Zadorov each had a goal and an assist, and Cale Makar had three assists for the Avalanche, the No. 2 seed in the West. Pavel Francouz made 33 saves.
Colorado center Nathan MacKinnon had two assists and leads the NHL with 20 points (seven goals, 13 assists) in the postseason. He has scored at least one point in all 11 games.
Game 4 of the best-of-7 series, scheduled for Friday, was
postponed
and will be played Sunday (6 p.m. ET; NBCSN, SN1, SN360, TVAS) in Edmonton, the West hub city.
"For us, it's gut-check time," Makar said. "Obviously, it's not how we drew it up going into the third period like that. We get some unfortunate bad breaks. I think we just stuck to our game, we kind of trusted that it would pull through. … I know everybody believes in this group; it's just about being able to go out there and execute."
WATCH: [All Avalanche vs. Stars Game 3 highlights]
Jamie Benn had a goal and an assist, and Anton Khudobin made 26 saves for the Stars, the No. 3 seed. It was their first loss in six games.
"Listen, we went into this, there's no chance we thought we were going to sweep this club," Dallas coach Rick Bowness said. "It's a very good team. We thought it'd be a six-, seven-game series. Hopefully, we don't get there, but if we get there, we get there. We're not going to dominate that club, they're too good, too talented."
Denis Gurianov scored to get the Stars within 3-2 at 4:42 of the third.
Blake Comeau, who played three seasons for the Avalanche from 2015-18, one-timed a loose puck near the top of the right face-off circle to tie it 3-3 at 9:02.
Benn deflected a shot by Esa Lindell at 10:47 to put Dallas ahead 4-3.
"Pretty crazy," Benn said. "Obviously being down two goals going into the third, you got to make a push, and I thought we had a great one. Back and forth, the ups and downs of the playoffs, we had them on the ropes. They're a good team; they came back, scored a couple of quick ones, and that was that. I liked our push, but we need it for 60 minutes."
Avalanche coach Jared Bednar called a timeout after Comeau tied the game, but the Stars look the lead 1:45 later on Benn's goal.
"Did they show me something after that? I would say yes," Bednar said. "… Today, I could feel it on the bench we were still in the game, still playing, needing something good to happen, needing to get rewarded, and [MacKinnon] and [Rantanen] end up making a play (on the tying goal), kind of giving us life again, and we had a good strong finish to the game. So kudos to our guys for giving it a full 60 or more if that's what it takes."