Marian Gaborik and Dustin Brown scored for Los Angeles (22-17-4), Anze Kopitar had three assists, and Peter Budaj made 26 saves for his 20th win.
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Brown tied it 2-2 with 5:08 left in the third period.
"We hit a lot of posts and we had a lot of opportunities and scrambles," Muzzin said. "We just capitalized on one and that got us to a tie and put us into overtime."
Drew Stafford and Dustin Byfuglien each scored a power-play goal, and Michael Hutchinson made 35 saves for the Jets (20-22-4). Mathieu Perreault had two assists.
"I really liked a lot of what we did and looked to build and got stronger and stronger," Winnipeg coach Paul Maurice said. "We needed the third one to go to put a team like that out."
Maurice was particularly impressed with Hutchinson and his play one night after he made 22 saves in 47:03 in relief of Connor Hellebuyck in a 4-3 loss to the Arizona Coyotes.
"A solid game," Maurice said. "Really, really good and battled around the net. A good confidence booster for him and our team in front of him."
Byfuglien gave Winnipeg a 2-1 lead 7:38 into the third with a point shot through traffic.
Gaborik scored 3:19 into the first period to give Los Angeles a 1-0 lead, taking a feed from Kopitar and banking it off the crossbar behind Hutchinson. It was Gaborik's third goal in 21 games.
Stafford tied it 1-1 at 15:06 of the second period. Perrault feed Stafford at the right faceoff dot, where he scored with a wrist shot.