The Blues also get a fourth-round pick in 2020 and forward prospect Erik Foley. According to reports, St. Louis retains half of Stastny's salary this season.
Winnipeg coach Paul Maurice said that the team was very happy to get a veteran center.
"They're just so hard to find," he said.
Maurice added: "(He's) a player that fits our game and how we want to play."
Stastny, who carries a cap hit of $7 million, will be an unrestricted free agent after this season. The Blues explored re-signing the 32-year-old, but the sides didn't reach agreement.
Winnipeg is second in the Central Division with 83 points, while St. Louis has slid to being one point out of the second wild card.
Stastny has 12 goals and 28 assists this year after going for 18 and 22 last year. The son of Hockey Hall of Famer Peter Stastny adds center depth behind Mark Scheifele and is solid at the faceoff dot.
"What he brings to the team is something that's going to be very helpful to us. Can play a lot of roles," Jets captain Blake Wheeler said. "Awesome faceoff guy. Great guy in the locker room,"