At the other end of the rink, Steve Mason made 31 saves for his first shutout as a member of the Jets.
It was his first NHL start since Jan. 9, after the 29-year-old worked his way back from his second concussion of the season.
"You're not going on the road all the time, so you're spending a lot of time at home. So it's nice to be back around the team and go on a road trip at an important point of the year," said Mason. "I felt really comfortable right from the get go. I had a good morning skate here this morning. I felt sharp right from the start, so it was a pretty good feeling for myself to get back in there after such a long lay off."
A patented Laine wrist shot got the Jets on the board first with 55.7 seconds to go in the opening period.
The shot - which came after a dominant shift from the trio of Laine, Stastny, and Nikolaj Ehlers - was made possible earlier in the shift when Dustin Byfuglien kept the offensive zone time alive with a timely pinch. He then went cross-ice to Stastny, who left it for Laine at the top of the left wing circle. The 19-year-old did the rest, ripping his 36th of the season short side on Henrik Lundqvist.
"To score with under a minute left, go into the dressing room with some momentum, I think that just gave us a little boost," said captain Blake Wheeler, who had an assist in the victory. "We've known how special he is - that shot, that release, it just looks like him Stastny, and (Ehlers), they have such a good chemistry. It seems like once they got together, Patty has had a lot more time to get that shot off."