SAN JOSE, Calif. — From last in the league in early January all the way to the brink of their first Stanley Cup finals in nearly a half century, it has been quite a ride for the St. Louis Blues.

A remarkable turnaround continued Sunday with perhaps their most convincing win of a charmed playoff run.

Jaden Schwartz started a dominant second period with the first of his three goals, Vladimir Tarasenko scored on a penalty shot, and the Blues moved within one win of the Cup finals with a 5-0 victory over the San Jose Sharks in Game 5 of the Western Conference finals.

“It’s probably tough to put into words,” Schwartz said. “It’s something that everyone’s worked for and dreamed about. You don’t want to look too far ahead. We all know how important and how hard that last win’s going to be. It would be a dream come true.”

St. Louis used a relentless forecheck to take control of the game and series in the second period, scoring twice and outshooting the beleaguered Sharks 20-6 during the period.

The Blues got 21 saves from Jordan Binnington in his first playoff shutout and a first-period goal from Oskar Sundqvist as they overwhelmed the Sharks.

Schwartz added two goals in the third for his second hat trick this postseason, becoming the first player with two hat tricks in one playoff run since Johan Franzen for Detroit in 2008.

The victory gave the Blues a 3-2 series lead, the closest they’ve been to making the finals since getting there in their first three seasons as the winner of the all-expansion Western Conference. St. Louis can earn its first trip to the finals since 1970 with a win at home in Game 6 on Tuesday night – an improbable journey for a team that was last in the NHL standings on Jan. 2.

“We’re close. We’re very close right now,” forward Patrick Maroon said. “I think the guys know that. It’s in the back of their heads, but we know that that’s a good hockey team over there, too, and they’re not going to give up.”

Martin Jones made 35 saves for the Sharks but got little help from his teammates, who have been held to one goal in losing the last two games. San Jose faces elimination for the third straight series, having overcome a 3-1 series deficit to Vegas in the first round and winning a Game 7 at home in the second round against Colorado.

“We’ve been here before,” Sharks Coach Peter DeBoer said. “Had to go on the road and win in Vegas in order to get to a Game 7. You’re never comfortable when your back’s against the wall like that, but we have been here before and found a way, and I’m confident we can do that again.”

San Jose got off to a spirited start in the rare afternoon contest that led to a more subdued crowd than usual. Evander Kane hit the post just 12 seconds into the game, and the Sharks had the better of the play in the opening 20 minutes despite falling behind 1-0 when Sundqvist converted a turnover from a hobbled Erik Karlsson into a goal less than six minutes into the contest.

Karlsson has been hampered by a groin injury that sidelined him for 27 of the final 33 games in the regular season, and he has been extremely limited since the third period of Game 4. He tried to make a quick outlet under pressure, but his pass went through teammate Brenden Dillon’s skates, off the boards and right to Sundqvist, who beat Jones to give the Blues their fourth goal of the series from the fourth line.

The Blues then took over in the second period, putting 11 shots on goal in less than five minutes. They added to the lead after Tarasenko’s shot was partially blocked. Jones swept it away, but it went right to Schwartz, who knocked it into the open net.

Schwartz’s two goals in the third period gave him 12 goals in the playoffs after scoring just 11 in 69 regular-season games.

“It was an off-year, obviously, for him in the regular season,” Coach Craig Berube said. “But the guy keeps working hard. He’s not going to change his attitude. He keeps with it, he keeps working, and it’s paying off now.”

The Blues kept up the pressure, leading to a breakaway by Tarasenko. He was pulled down by Brent Burns for a penalty shot and converted it with a shot high to Jones’ glove side.

The game got out of hand in the third as the Sharks took a parade to the penalty box, upset about a hit to the head of Tomas Hertl in the first period by Ivan Barbashev and a high hit to captain Joe Pavelski by Alex Pietrangelo early in the third.

“I thought we obviously took way too many penalties,” forward Logan Couture said. “You can’t come back when you’re in the box the whole period; (it) got away from us at the end. Would have really liked us to control our emotions and given ourselves a chance.”

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