Hutton's NHL career high in saves led to his first shutout of the season and ninth in the League. He made 36 saves in the first two periods, 21 in the second.
"He was 48 saves good," St. Louis coach Mike Yeo said. "He was really good. I thought our third period we tightened up a little bit, but obviously, as tight as we want to be, that team's going to find a way to get chances. They've got a lot of skill, a lot of firepower. But we gave up a little bit off the rush tonight, but every time we gave up something, [Hutton] was there."
Vladimir Tarasenko and Vince Dunn scored power-play goals for St. Louis (22-10-2), which went 2-2-0 on a four-game homestand.
"I always told you guys [Hutton's] a really good goalie, same as Jake [Allen], and we are really thankful for those goalies," Tarasenko said. "We're really proud how [they're playing] right now."
Steve Mason made 28 saves for the Jets (18-10-5). It was his first game since Nov. 25, when he was hit in the head with a puck. It was the second time Winnipeg was shut out this season.
The Jets are 0-4-2 their past six road games and have been outscored 24-10. Winnipeg hosts St. Louis on Sunday.
"That was a [heck of a] road game tonight, wouldn't you say?" Wheeler said. "We played pretty good sometimes; that's how it works, and I wouldn't call them [road] woes. I would just say we are really [darn] good at home (11-3-1)."
Tarasenko's 15th goal gave St. Louis a 1-0 lead at 9:16 of the first period. He followed Alexander Steen's slap shot from the right circle and got to the rebound.
Dunn's goal with 2:26 left in the third period, a wrist shot from inside the blue line, gave St. Louis a 2-0 lead.