The Jets won their second in a row and kept the Panthers stuck on seven victories this season, fewest in the NHL.
Byfuglien, without a goal in his last 13 games, broke past two defenders in the slot and sent a wrist shot past the glove of Jacob Markstrom.
Winnipeg let a two-goal lead get away before coming through in the extra period.
Andrew Ladd and Blake Wheeler scored in the opening period for the Jets, but Jonathan Huberdeau tipped in a shot with 13 seconds left in the first stanza to bring the Panthers within one. It was Huberdeau's 12th of the year.
Shawn Matthias tied it 5:50 into the third period.
The win went to Ondrej Pavelec, who made 38 saves. Markstrom allowed the three goals on 28 shots.