Winnipeg trails the Nashville Predators by two points for first place in the Central Division.
Jake Allen made 19 saves and lost for the 12th time in the past 14 starts for the Blues (34-24-4), who have lost a season-high five in a row.
"If I had the answer it wouldn't be happening," Blues captain Alex Pietrangelo said of the recent struggles. "... We've just got to show up, enough talking. You can only say so much; you've got to go out and prove it."
St. Louis holds the first wild card into the Stanley Cup Playoffs from the Western Conference, one point ahead of the Los Angeles Kings and Calgary Flames.
"It's cliche, we've talked about it, but it's got to start showing up on the ice," Blues forward Alexander Steen said. "… Twenty games [remaining] and we're in a fight here."
Nic Petan and Wheeler scored 45 seconds apart in the first period to give the Jets a 2-0 lead.
Petan scored at 10:03 after Blues defenseman Chris Butler's giveaway for a 1-0 lead, and Wheeler scored from the right circle at 10:48 to make it 2-0.
Connor's fourth goal in four games, on the power play, increased the lead to 3-0 at 18:31.
Patrik Laine's 29th goal of the season and fourth in as many games, a wrist shot from the right circle at 6:04 of the second period, made it 4-0 on the Jets' 14th shot.