Alex Steen won a battle for the puck along the end boards and got it to Tyler Bozak who fired a centering pass which Schwartz hit past a shocked Connor Hellebuyck, who could only manage to get a piece of before the mesh bulged.
"I think their line (forwards) and their D were a little bit tired from the lines before," he said. "Just a good grinding shift, we kept the puck below the goal (line) and ended up winning it back a couple of times. We were kinda relentless on the puck."
"(Tyler Bozak) came on, he was fresh, and probably knew there wasn't much time left so he just threw (the puck) on net and I kind of got lucky, it just hit my stick," said Schwartz of his tally, which put St. Louis within a win of advancing to the Western Conference Semi-Finals.
The goal capped a three goal comeback by the Blues, who scored all of their goals in the third period after trailing 2-0 after two periods.
Ryan O'Reilly and Brayden Schenn also scored for St. Louis.
Jordan Binnington turned aside 29 shots.
Adam Lowry and Kevin Hayes scored for Winnipeg.
Connor Hellebuyck made 26 saves for the Jets.
Adam Lowry gave the Jets the lead 12 seconds into the contest when he lifted a rebound off a Brandon Tanev shot from the corner past Blues goaltender Jordan Binnington to make it 1-0.
Kevin Hayes made it 2-0 13:56 into the first when he took a pass from Dustin Byfuglien before driving the net on his off wing and stuffed his own rebound through Binnington's pads.
The goal was his 2nd of the playoffs.
The Jets centreman, acquired at the NHL trade dealine, had a strong second period, narrowly missing giving the Jets a 3-0 lead 12:52 into the period when his second effort to poke the puck past the goal line actually sent it in the other direction.
"The puck went through his legs, Parayko lifted my stick and my stick came down and hit it out," said Hayes.
"We had a pretty good second half of the game and I thought our best period was the third. They scored three goals," he said.
“We’ve been down before and I’m sure the guys will come back and regroup, watch some film and be ready for Game 6," added Hayes.
There was no scoring in the second period.
St. Louis got on the board 1:29 into the third period when Ryan O'Reilly snapped the disc past Hellebuyck off a rebound from a Brayden Schenn shot.
The goal was his 2nd of the playoffs.
Brayden Schenn tied the game on a controversial goal at 13:52 when he buried a backhander past Hellebuyck as the net was knocked off it's moorings.
Oskar Sundqvist hit Schenn with the pass but was knocked into the net by Jets defenceman Dustin Byfuglien as the puck was entering the goal.
Officials ruled the net was knocked off by a Jet player.
The goal was his first of the playoffs.
Blues Head Coach Craig Berube credited Sundqvist with getting the inside track on Byfuglien.
"He did a great job driving to the net, took the puck wide. He’s been a real good player for us for the whole season. Playing wing, middle, doesn’t matter. He does a great job for us killing penalties. He played against Scheifele’s line all night, thought he did a great job.”
Jets Head Coach Paul Maurice said the game was like it has been for the entire series, tight.
"There’s too much good action. So it’s almost been a low event series, not a tremendous number of shots, there’s not 45 shots. But everything is dangerous. There’s going to be chances that you’re shaking the head at, Kevin Hayes alone had a hat trick on his stick. There’s going to be a bunch of those. It’s tight. It’s going to be one more save at the end."
Blues defenceman Alex Pietrangelo said the series has made little sense with the home team yet to win a game.
"You don't see that too often right? You don't see the road team win all the games but, again, we've got to go home and we played better in Game 4 there so we've got to build off that," he said.
Jets captain Blake Wheeler was succinct in his assessment of the game.
"No different than being down 0-2. Happens all the time, man. Gotta win a hockey game. Won almost a hundred of them the last two years. So we're confident we can do that," he said.
Game 6 is Saturday night in St. Louis.
SHOTS-ST. LOUIS 29 WINNIPEG 31