The Flyers almost won it 2:13 into overtime, but it was ruled that Briere kicked the puck into the net on a power play.
Van Riemsdyk, ahead of the pack, was denied by Marty Brodeur with 3:15 to go in regulation.
Trailing 3-2, New Jersey scored the equalizer as veteran winger Petr Sykora scored from the left circle after faking a slap shot, then ripping a drive between the legs of goalie Ilya Bryzgalov with 8:38 left in regulation.
Jakub Voracek turned the puck over to David Clarkson, who fed Sykora for the goal, which was scored shortly after a timeout by Flyers coach Peter Laviolette.
About six minutes earlier, the Flyers had taken a 3-2 lead as Claude Giroux one-timed a slap shot from the left circle past Brodeur.
Until that power play, the Flyers had been 0-for-3 and had managed just one shot on three man-advantages.
Game 2 is Tuesday night at the Wells Fargo Center.
The Devils took the game's first 11 shots, but the Flyers regained their legs and started to take control after a New Jersey-dominated first period.
Goals by Briere (breakaway) and van Riemsdyk 37 seconds apart gave the Flyers a 2-1 lead in the second period. But a questionable penalty on Matt Carle led to a power-play goal by Travis Zajac and sent the teams into the third period tied at 2-2.
Briere, sent in alone on a pass by Voracek, beat Brodeur with a forehand shot and knotted the game at 1-1 with 11:53 remaining in the second period. It was his sixth goal in seven playoff games this spring.
Shortly thereafter, Brodeur made a bad clearing pass and had to make a sprawling save on Erik Gustafsson, but van Riemsdyk knocked in the rebound to give the Flyers their first lead.
While the Flyers were on a power play, Devils rookie Adam Henrique broke in alone with Carle in pursuit. Carle appeared to barely touch him from behind, but a hooking penalty was called.
New Jersey capitalized. Bryzgalov failed to control a wraparound pass behind his net, and the puck went to Zach Parise, who crossed it to Zajac for the tap-in that went off the goalie and trickled past the goal line with 6:07 left in the second period.
Between the first and second periods, the Wells Fargo Center scoreboard showed contestants in a deviled egg-eating contest at Xfinity Live!
One fan's comment was picked up on the microphone: "That's so (expletive deleted) gross!"
That's probably what Laviolette was thinking about the first period.
The Devils swarmed Bryzgalov from the start, took an early 1-0 lead and outshot the Flyers, 15-6, in the opening period. The Flyers' power play, which was a staggering 12 for 23 against Pittsburgh, failed to get a shot on a pair of two-minute advantages in the first period.
After having six days between games, the Flyers preached that they had to come out strong because the Devils would still have a "rush" from their Game 7 double-overtime win in Florida on Thursday.
They were correct on the latter.
A turnover by Matt Read led to the Devils' first goal, scored by Parise on a tracer from deep inside the left circle just 3:16 into the game. Parise took a feed from Patrik Elias, who intercepted Read's pass behind the net that was intended for Carle.
That gave the Devils as many goals in 3:16 as they had scored in 3 1/2 games against Bryzgalov during the regular season.
Bryzgalov was strong in the period, however, and kept the Flyers within striking distance.
The Flyers were coming off a six-game victory over Pittsburgh, the Stanley Cup favorite. The Devils won overtime decisions in Games 6 and 7 and get past Florida in their conference quarterfinal.
Contact Sam Carchidi at firstname.lastname@example.org, or on Twitter.com.