The Blues scored three goals against Connor Hellebuyck in a span of 3:08 late in the third period to pull the game to a 4-4 tie.
"We had some good stretches, we had some tough stretches, and I made a couple mistakes tonight," Hellebuyck said. "But we got the two points, and that's all that really matters."
Anthony Mantha had two goals and an assist for the Devils (11-5-0), who have won three straight and are 7-1-0 in their past eight games.
Bratt and Luke Hughes each had a goal and an assist, Jack Hughes and Brett Pesce each had two assists, and Hellebuyck made 28 saves in his fourth straight win since being acquired in a trade with the Winnipeg Jets on Oct. 23.
"These moments are going to define a season," Hellebuyck said. "This will probably be forgotten in a week but the lesson learned from it won't be, right? It's just continuing to build our game. These little things mean a lot to us, and they kind of get forgotten later on, but they're important."
Philip Broberg and Mason McTavish each had a goal and an assist, and Binnington made 31 saves for the Blues (7-7-2), who have lost two straight (0-1-1).
McTavish picked up the rebound of Justin Barron's shot and scored from the slot to pull St. Louis within 4-2 at 11:46 of the third period.
"We showed a lot of heart sticking with it," St. Louis captain Robert Thomas said. "I thought in the second period we did a lot of good things ... they obviously scored two quick goals, but it was a tight game and we got rewarded in the third. They all came in a bunch there, so it was nice to get a point, but we would have liked to have finished it off."
Broberg then made it 4-3 on a wrist shot from the left face-off circle at 14:36, and Dalibor Dvorsky tied the game at 4-4 just 18 seconds later off a tip-in from the slot at 14:54 when Connor McMichael stole an attempted pass by Hellebuyck in the right circle.
"I'll just call it a mistake, how about that?" Hellebuyck said.