Brossoit denied Hall while sliding to his left off a 2-on-0 break with Hischier 45 seconds into overtime.
"You know what, sometimes I don't mind a 2-on-0," Brossoit said. "A lot of the times the opposing players will complicate things rather than just a breakaway where they'll just shoot it on goal. I was lucky I was able to come across to make the save."
Morrissey scored at 10:32 of the third period, and Copp at 13:21 to give the Jets a 3-1 lead before Bratt and Hall scored in a span of 2:23.
"We were coming in back-to-back versus a rested team," Devils coach John Hynes said of New Jersey, which lost 6-3 at the Washington Capitals on Friday. "I really liked the way our team played. Even though we got down when we gave up a couple of goals in the third, to get two goals in under five minutes in the third, we worked our way into a point."
Bratt pulled the Devils within 3-2 on a power-play goal at 15:41, and Hall tied the game after a scramble in the crease with 1:56 remaining and Kinkaid pulled for an extra attacker.