Connor McDavid had two assists, and Stuart Skinner made 19 saves for the Oilers (37-24-4), who are 3-8-0 in their past 11 games. Edmonton next plays at the New York Islanders on Friday.
"I thought our guys put in a really good effort the first 40 minutes; it was an even game," Edmonton coach Kris Knoblauch said. "In the third period, I thought we pushed hard and had a lot of good scoring chances. I think our top line (Leon Draisaitl, McDavid, Zach Hyman) controlled the play when they were on the ice."
Bratt tied it 2-2 at 6:50 of the third period with a wrist shot from the high slot.
Simon Nemec then put New Jersey ahead 3-2 with his first goal of the season on a slap shot from the point at 8:25.
"I felt like, finally," Nemec said. "I was waiting for this moment all season. It was a great game for us and it was game-winning goal, so it was more special."
Pesce gave the Devils a 1-0 lead at 3:58 of the second period with a snap shot from just outside the right post after Bratt’s one-timer from the slot was knocked down in front.
Draisaitl tied it 1-1 at 10:05 with his League-leading 47th goal of the season on a wrist shot from the right face-off circle. He extended his point streak to 16 games (12 goals, 12 assists).