Bratt had a goal and an assist, and Nico Hischier and Curtis Lazar scored for the Devils (5-4-2), who have lost four straight (0-2-2) and were coming off a 5-3 loss at the Detroit Red Wings on Thursday. Allen made 26 saves.
"Today we were chasing the game right from the get-go and that's a challenge, but guys stayed with it until the end, and I thought we got better as the game went on and we sort of adjusted back to the way this game was being played," New Jersey coach Sheldon Keefe said. "Listen, the League is hard. Tonight was our ninth game in the last 16 days. It's a tough schedule ... that's almost unheard of.
"We looked like a team who hasn't practiced much; we've had two practices in the last 16 days. All things considered, to see our team fight on the back to back, third game in four nights and make sure to get the point, those are all really encouraging."
Nelson gave New York a 1-0 lead at 1:23 of the first period, tipping a shot from the left face-off circle over the left shoulder of Allen. Defenseman Adam Pelech took the initial shot from the left point.
Hischier tied it 1-1 at 11:36 on a snap shot from the slot, his fifth goal in the past four games.
"Obviously, missing out on the two points is hard but we did a good job in fighting back to get into a position of getting one point," Bratt said. "I mean, towards the end of the season, that one point can make a big difference."
Anders Lee gave the Islanders a 2-1 lead at 14:03, scoring a power-play goal off a rebound outside the left post.