Each team scored three times in the first period.
Palmieri, who has four goals in his past two games, gave the Devils a 1-0 lead at 1:38.
Egor Yakovlev shot wide and the puck went off the end boards and came out just to the left of Miller. Palmieri was in position to lift the puck over the outstretched left pad of Miller for a 1-0 lead.
Jakob Silfverberg scored for the third straight game to make it 1-1 at 2:12.
Anaheim defenseman Josh Manson stopped the puck just before it exited the New Jersey zone and passed it to Silfverberg, who made a stick move to turn around Devils forward Nico Hischier and scored from just above the left circle.
Kiefer Sherwood gave Anaheim a 2-1 lead at 2:47. He tried to center a pass to Carter Rowney but the puck hit the stick of New Jersey forward Stefan Noesen, went off his leg and across the goal line for a 2-1 lead at 2:47.
It was the fastest three goals by two teams to open a game this season.
Palmieri made it 2-2 at 10:49. Taylor Hall stole the puck behind the Anaheim net, fed Palmieri from below the goal line and he scored on a one-timer from the right circle.
New Jersey regained a 3-2 lead when Damon Severson scored on the power play at 13:08.
After Montour was called for hooking Travis Zajac at 11:46, Johansson received a pass just below the goal line and he made a diagonal pass through the crease to Severson, who scored on a one-timer.
Anaheim was headed for its first power play but Montour scored from the right circle before it could begin, tying the game 3-3 at 13:39.
"That first period was a little helter-skelter," Schneider said.