Hossa scored with 55 seconds left to lift Detroit to a 3-2 win over Minnesota on Sunday, snapping the Red Wings' three-game losing streak.
Chris Osgood made 20 saves.
"We knew how desperate they are. They need to win every game to get into the playoffs," Hossa said. "We felt an urgency in the dressing room, too, that this was a playoff game."
Red Wings coach Mike Babcock and former Detroit coach Scotty Bowman are the only coaches to win 50 games in four consecutive season.
Niklas Backstrom made 41 saves, Cal Clutterbuck had a goal and an assist and Marian Gaborik also scored for the Wild, who are 10th in the Western Conference and needed at least a point to retain a realistic chance of reaching the playoffs.
Minnesota was less than a minute left from overtime and getting that point when Hossa broke their hearts.
"We came here to play good and try and take two points," Backstrom said. "We came close to getting one, but we didn't come out with anything."
Gaborik, who has seven goals and 10 points in eight games since returning from groin problems that forced him to miss 65 games this season, tied it with 2:03 left in the second period when he scored his 10th goal from the slot during a goal-mouth scramble.
Hossa scored his 38th by beating Backstrom from the right circle after taking a pass from Holmstrom.
"Pasha [Datsyuk] got the puck and he just chipped it to Homer [Holmstrom], and Homer took the puck to the middle and he just serve it to me perfectly," Hossa said. "I just tried to make sure the puck was stable on my stick and tried to shoot it high glove and it worked."
That left the Wild mulling what could have been. They're three points out of the final playoff spot with three games remaining.
"The only thing we can do is work as hard as we can," Clutterbuck said. "The rest is out of our hands."
Detroit is trying to get dialed in as the playoffs approach.
"I think the third period was the kind the kind of hockey you're going to have to play come playoff time," Nicklas Lidstrom said.
Detroit led 2-1 after a wild first period.
Hudler opened the scoring with 8:39 left with his 23rd goal, putting in a one-timer from the slot off a pass from Henrik Zetterberg, and Datsyuk made it 2-0 just 1:04 later when he moved across the slot and finally backhanded a high shot over a sprawling Backstrom.
Minnesota took advantage of Osgood's gaffe to get on the board with 1:26 left in the opening period. Clutterbuck intercepted the goalie's pass along the right wing boards and put a shot through as Osgood scrambled to get back into position.
Detroit D Brian Rafalski missed his third game with a sore groin. ... Red Wings' C Tomas Kopecky returned after missing a game with a sore back. ... Clutterbuck, one of the league's best trash talkers, was wired for sound for the nationally televised game by NBC. ... Minnesota D Nick Schultz left the game in the second period with an eye injury. Coach Jacques Lemaire said the team didn't know how bad it was or if Schultz would be able to return this season.