Schwartz scored with 15 seconds remaining to cap St. Louis' three-goal third period in a 3-2 win over the Jets on Thursday night, giving the Blues a 3-2 lead in their Western Conference playoff series.
"(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," Schwartz said of his first goal in the playoffs.
"If we lose, we're done," Hayes said. "That's enough motivation. Game 6 will be our best game."
The Blues' win marked the first time in 15 years, and fourth time overall, the road team has won each of the first five games in a seven-game series.
"You don't see that too often, right?" Blues captain Alex Pietrangelo said. "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."
According to a league stat, when a best-of-seven NHL playoff series is tied 2-2, the winner of Game 5 takes the series 78.8 per cent all time (205-55).
St. Louis went on its first power play with 21 seconds remaining in the second period when Jacob Trouba was called for roughing. The period ended with Winnipeg holding the 22-21 edge in shots on goal.
O'Reilly scored with 11 seconds remaining in the power play, off a rebound in a crowd in front of the net at 1:29 of the third to pull St. Louis to 2-1.
Schenn tied it with 6:09 remaining following a review. Oskar Sundqvist passed the puck over to Schenn and then was pushed down by Jets defenseman Dustin Byfuglien and into the net, knocking it off. A review determined Schenn had shot the puck before a defensive player caused the net to come off.
That set up Schwartz's winner in the final minute.
"Lucky pinballs," Hellebuyck said of the Blues' go-ahead goal. "The puck just bounced and ended up right on their tape. Tough to eat that one, but I thought we were the better team. If we keep fighting here, it's not over."
Winnipeg mounted a 2-0 lead after the first period, even though the Blues had the 15-12 advantage for shots on goal.
Lowry got the crowd fired up 12 seconds into the game when he used Brandon Tanev's rebound to notch his first goal in the series. It set a franchise record for fastest goal in the playoffs. Trouba had the previous mark of 31 seconds, scored in Game 5 against Minnesota last season.
After fans stopped cheering, some on the rink side of St. Louis's net started chanting "You look nervous" to Binnington. Before Game 1, a video clip of the rookie saying, "Do I look nervous?" to reporters made the rounds on social media.
"Yeah, that's incredible," Binnington said of hearing the chants. "You get what you ask for because when I was younger, I really enjoyed that stuff and always wanted it, and now it's here. It doesn't feel as good as I thought, but it's cool, it's really cool."
Hayes cut to the front of the net with 6:25 left in the first and sent the puck under Binnington's pad to make it 2-0 with his second goal of the playoffs.
Hayes almost scored twice more in the second period.
He took away his own goal when his close shot slid under Binnington and started going toward the goal line. He tried to swipe it across, but his stick pulled the puck out instead.
Hayes also had a breakaway late in the period, but Binnington sprawled on the ice and the shot from the side went into his pads.
Winnipeg got a four-minute power play when Robert Thomas was dinged for high sticking Mark Scheifele in the face shield, but the Jets only had one shot on goal after a number of misses and blocked attempts.
Another Winnipeg power play early in the third delivered one shot on goal.
The last series in which the road team won the first five games was in 2004 when Calgary beat San Jose in six games, clinching it in Game 6 at home. ... Schwartz's goal was the second-latest winning goal in regulation in franchise postseason history. Only Gino Cavallini's goal with 9 seconds left in Game 3 of the 1990 division finals against Chicago came later. ... St. Louis was missing injured third-pairing D Robert Bortuzzo, who was replaced by Carl Gunnarsson. ... Lowry's goal was the fifth scored in the opening minute this postseason, tied for the most in the opening round of any playoffs. The others were in 1981, 2012, 2016.
Game 6 is Saturday night at St. Louis.
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