Gionta, first up for the Devils in the shootout, whipped the puck past Antero Niittymaki to cap New Jersey's comeback.
Simon Gagne, first for the Flyers in the shootout, fired wide and Martin Brodeur made saves on Peter Forsberg and Mike Richards.
Scott Gomez and Jamie Langenbrunner also scored for the Devils in regulation. New Jersey has won 10 of its last 13 and enters the all-star break with a 13-point lead atop the Atlantic Division.
Ryan Potulny, Kyle Calder and Gagne scored for the Flyers, losers of eight straight.
With Brodeur on the bench for an extra attacker, the Devils threw everything at Niittymaki in the closing seconds. Niittymaki turned back Jamie Langenbrunner, Zach Parise and Paul Martin but was helpless as Martin took another shot and Gionta knocked in the rebound for his team-leading 21st goal.
Gagne staked the Flyers a 3-2 lead on a power-play goal with 6:58 remaining in the third. Mike Knuble passed the puck to Gagne who had positioned himself along side the Devils net. He took one stride and lifted the puck over Brodeur.
Potulny opened the scoring 2:25 into the game as he used Devils defenceman Brad Lukowich as a screen to beat Brodeur from the high slot with an unassisted goal.
A screen also figured prominently in Gomez' score with 4:35 left in the period. Gionta jumped into Niittymaki's line of sight just Gomez unloaded a drive from the top of the right circle for a power-play goal.
The Devils outshot the Flyers 11-6 in the first period.
New Jersey also dominated the shot count, 12-6, in the scoreless second period. One of New Jersey's best chances in the period didn't register in the total as Eric Rasmussen lifted an inviting rebound over the Flyers' net.
In the third, Calder gave the Flyers a 2-1 lead at 1:58 on a goal created by hard work by Jeff Carter behind the Devils net. Carter wrestled the puck away from New Jersey defenceman Andy Greene and wheeled a pass across the slot to a wide-open Calder.
Langenbrunner evened the score at 2 at 7:58 with a goal that came 4 seconds after a Devils power play had expired. Greene fired a shot from the point and Langenbrunner knocked in the rebound with Greene getting his first NHL point.
Notes: With D Johnny Oduya having recovered from a shoulder injury, D David Hale was a healthy scratch for a second straight game. ... LW Geoff Sanderson sat out for the Flyers. He returned in the Philadelphia's previous game after missing 16 in a row with a hernia. ... The Devils have won seven straight from the Flyers. ... Niittymaki's personal losing streak stretched to 14.