Chris Wagner and Patrice Bergeron scored, and Jaroslav Halak made 28 saves for the Bruins (20-14-4), who have lost two in a row after a three-game winning streak.
Severson gave the Devils a 1-0 lead with a goal from the right face-off dot 25 seconds into the first period.
Palmieri scored a power-play goal at 15:25 to extend the lead to 2-0. Halak saved Palmieri's slap shot but the rebound went off defenseman Brandon Carlo's leg into the net.
"Just give them credit, they came out played their game much better than us early on," Bruins coach Bruce Cassidy said. "Having said that, I do believe the second goal gave them a lot of juice too. It's a fortunate goal, goes in off Carlo, not much happening and they've got a 2-0 lead. Jake [DeBrusk] hits a crossbar, then a second one hits a post. A break here or there the game probably evens out and then who knows where it goes from there?
"But we're chasing it the whole night, and when you chase it you usually end up not catching it. That's just the way it works in this league."
