That souvenir now belongs to his 10-day-old daughter, Lyla Grace, who was born March 17.
"It's definitely going to her," he said.
St. Louis has won three in a row overall and nine straight against the Wild. The Blues lead the NHL with 107 points.
He has a 2.03 goals-against average and a .918 save percentage with the Blues.
Zach Parise scored for the Wild, who had earned points in their previous three road games but have lost six of eight overall. Darcy Kuemper made 19 saves but fell to 12-8-4. He has won just twice in nine starts.
Oshie, who has a career-high 21 goals, scored twice in the first period -- including a short-handed goal with 26 seconds left that made it 2-0.
He converted from the slot at 4:43 and then beat Kuemper on a breakaway for the Blues' fifth short-handed goal of the season.
Oshie tipped in a shot from Jay Bouwmeester at 7:18 of the third period to complete the hat trick.
"This has been an exciting couple of months for me," said Oshie, who played for the United States in the Olympics, "but she trumps everything for me."
The modest Oshie refused to take credit for his big night.
"The pucks just happened to go off my stick," he said.
"He's been a horse for us all year," Backes said. "He strung a couple of great shots together and a tip on the last one.
"I couldn't be happier for him."
The Blues posted a hat trick for the second straight game, as Backes scored three times in a 5-3 win Tuesday at Toronto. This is the first time St. Louis has recorded back-to-back three-goal games since Dec. 20-23, 2000, when Pavol Demitra and Michal Handzus accomplished the feat.
The Blues lead the Western Conference and are battling Boston for the Presidents' Trophy for the top mark in the NHL. They are one point ahead of the Bruins with nine games remaining for both teams.
"We wanted to make a statement to the league that we deserve to be where we're at," Oshie said. "We want that target on our backs."
St. Louis coach Ken Hitchcock believes his club is peaking at the right time.
"We're not going to be overwhelmed because we're the favorites in a playoff series," he said. "We're going to embrace it."
Even Minnesota coach Mike Yeo was impressed with the Blues.
"They've got a great team" he said. "They smelled blood and they went after it.
"This was a team loss."
Miller made nine saves in the first 6:07 before coasting to the win.
"We got settled into our 5-on-5 game and really limited the puck possession by the other team," he said. "This was a big step forward."
Schwartz netted his 23rd goal of the season on a power play midway through the second period to make it 3-0.
The Wild lost for the second time in two nights after dropping a 5-3 decision to Vancouver on Wednesday.
"I don't think anyone can really be happy," Kuemper said. "We've got to turn this thing around."
St. Louis has killed 23 straight power plays against Minnesota over six games.
Steve Cash and Josh Pauls, members of the USA's gold-medal-winning Paralympic hockey team, were honored before the game. ... Minnesota began a four-game trip that also has stops at Phoenix, Los Angeles and Chicago. ... The Blues are shooting for their first Presidents' Trophy since the 1999-2000 season. ... Minnesota LW Dany Heatley has gone a season-high 13 games without a goal. ... Wild RW Jason Pominville led all players with seven shots on goal.