Byfuglien beat Hurricanes netminder Justin Peters off a rebound from a shot by Andrew Ladd, who finished with three assists.
Carolina's Alexander Semin tied the game 3-3 with 1:12 left in the third period with Peters on the bench for an extra attacker, beating Jets goalie Ondrej Pavelec on the glove side.
Blake Wheeler, with a goal and two assists, Zach Bogosian, with a goal and an assist, and Paul Postma also scored for Jets (23-19-2), who remained ninth in the Eastern Conference but moved to within two points of the Southeast Division-leading Washington Capitals.
Washington lost 3-1 to Ottawa on Thursday and has 50 points and the coveted automatic spot in the East Conference playoffs.
The New York Rangers are tied on points with Winnipeg hold down the final playoff spot in the East with a game in hand after a 6-1 victory over Florida.
Winnipeg, meanwhile, has now won a season-high five straight.
Tuomo Ruutu scored Carolina's other goals, his first and second of the season coming in his 12th game with the Hurricanes (17-23-3). Jordan Staal had a pair of assists.
Pavelec made 24 stops, including one spectacular one in the second period that sparked the crowd at MTS Centre to give him a standing ovation.
Carolina defenceman Justin Faulk fired a wrist shot from close in at Pavelec and the netminder whipped out his stick and the puck slammed into his stick and back enough for Pavelec to smother it.
Peters finished with 27 saves.