The Devils overcame a 3-2 deficit in the series to win Game 7 and set up an Eastern Conference semifinal matchup with Philadelphia.
New Jersey let a 2-0 lead get away in the third period but Martin Brodeur made 11 of his 43 saves in the first overtime and came away with the 103rd playoff victory of his career.
The overtime goal was Henrique's second score of the game. He tipped in a shot by Anton Volchenkov 1:29 into the contest and Stephan Gionta scored in the second period for a 2-0 New Jersey lead.
Brodeur gave up third-period goals to Stephen Weiss on a slap shot and Marcel Goc on the power play. Goc's goal came with 3:28 to go in regulation, with Marek Zidlicky serving a 2-minute penalty for delay of game.
The Devils have reached the second round of the playoffs for the first time in five years and will take on a Philadelphia team that produced 30 goals while eliminating Pittsburgh in six games.
The setback brought an end to Florida's first playoff experience in 12 years.