T.J. Oshie scored the only goal in the shootout 3 victory over the Devils at Prudential Center.
Brian Elliot relieved an ineffective Jaroslav Halak in the second period and stopped all 15 shots he faced before denying three Devils skaters in the shootout.
Patrik Berglund scored with 5:50 remaining in regulation to send the game to overtime, while Chris Stewart and David Perron also scored for St. Louis, which ended its three-game road trip with two straight victories.
"Sometimes if you're playing good teams they are going to win too," said Hedberg. "They're a good team and they play consistent. They don't turn away from their system."
Elias and Ilya Kovalchuk executed a perfect give-and-go to chase Halak and give the Devils a 3-2 lead with 10:48 remaining in the second period.
The lead stood early in the third after New Jersey killed off Kurtis Foster's five-minute major for boarding, but a deft deflection later in the period resulted in a tie game.
Roman Polak flicked a wrister on net from the right point, and Berglund, staged in front of a Hedberg, raised his stick above his head and with a downward chop re-directed the shot into the back of the net. Replay was used to determine if Berglund's stick was above the crossbar.
"Whenever you're down one goal going into the third on the road it's a good position to be in, so we were pretty comfortable," said Elliot.
Henrique, who was denied on a breakaway early in the third, was stopped in close in the final minute of overtime, and Kovalchuk, Elias and Parise each came away empty in the shootout.
Parise scored on the power play in the first period when he took Kovalchuk's cross-ice pass and fired a wrister from the right circle that tipped off the top of Halak's glove and tricked in at the 16:46 mark.
Stewart backhanded one between Hedberg's pads two minutes later, and Perron scored 4:50 into the second on a shot similar to Parise's tally.
Henrique tied it again just 45 seconds later on a wrister from the left side.
St. Louis improved to 11-0-2 against the Eastern Conference...Parise and Elias both extended their point streaks to seven games...Halak made nine saves before getting pulled...The Blues had three shots in overtime to the Devils' two.