Fellow rookie Nick Palmieri tied it with 2:54 to play in regulation and Johan Hedberg made 20 saves and picked up an assist on the game-winner in helping the Devils win for the 10th time since Jan. 9 (10-1-2).
Rookie Jeff Skinner had two goals for Carolina, which moved a point ahead of idle Atlanta in the race for the eighth and final playoff berth in the Eastern Conference.
Brodeur hurt his right knee in a win Sunday over Montreal and the team listed his status as day to day.
Devils general manager Lou Lamoriello said the 38-year-old Brodeur will have to rehabilitate his knee before he can play again so it's unlikely that he will be back soon.
Still, the Devils have closed within 15 points of a playoff berth following a dreadful start to the season.
Tedenby's game-winner came on a counter attack. The Swede took a pass from Travis Zajac, skated between the circles and zipped a shot past Cam Ward for his seventh goal of the season.
Palmieri forced overtime by backhanding the rebound of Ilya Kovalchuk's shot. The Russian skated through the Hurricanes' defense to take the shot and set up Palmieri's fifth goal in 15 games.
New Jersey had three great chances to win in regulation, but Ward stopped a backhander by Jason Arnott with 2:12 left, made a glove on Andy Greene with 45 seconds to go and turned in a brilliant save in close on Patrik Elias with 14.4 seconds remaining.
Skinner, who also scored two goals in a 6-3 New Year's Day win over New Jersey, seemingly scored the winner at 8:37 of the third period with his second power-play goal of the game.
Just 6 seconds after Anton Volchenkov got his third minor penalty of the game - the only three by New Jersey - Skinner tallied his 20th of the season by tapping in the rebound of a shot by Eric Staal.
Hedberg is going to have to play well if the Devils want to continue their run with Brodeur out, and Skinner's first goal wasn't a good one from that standpoint.
Skinner broke into the Devils' zone after taking a pass from Joe Corvo and beat Hedberg with a shot from low in the left circle that seemed to graze off the goalie's jersey and into the net at 9:22 of the second. It was Carolina's first shot of the period.
Skinner's 43 points and 23 assists are tops among NHL rookies, and his 20 goals rank second.
Tedenby tied the game a little more than 2 minutes later after being set up right in front of Ward by a no-look pass from Arnott. The Hurricanes' goaltender never had a chance.
Troy Bodie, who hasn't scored in 43 games this season, had the best chance in the opening period when the rebound of Corvo's shot landed at his feet with the net open. However, the forward couldn't control the puck and the best he could do was slide a shot into Hedberg.
NOTES: Hurricanes forward Jussi Jokinen missed his third straight game with a lower-body injury. ... Ward made his 13th consecutive start. ... Corvo's two assists gave him 100 with the Hurricanes. ... The game was the opener of a five-game road trip for Carolina. The final game also will be played here next week. The teams will meet three times in 12 days.