Dawson Mercer pushed it to 3-1 when he poked in a loose puck to the right of Annunen at 6:47.
Colorado scored twice in 29 seconds to tie it during a 4-on-4 early in the third period.
Cale Makar pulled the Avalanche within 3-2 on a shot at the left post at 8:27, and Samuel Girard made it 3-3 with a slap shot from the top of the left face-off circle at 8:58.
"We started off slow but had a great third period," Colorado forward Miles Wood said. "That's the way it goes sometimes. Back-to-backs are tough. It comes with the territory of playing in the NHL."
Haula scored into an empty net with one second remaining for the 5-3 final.
"I think the push we gave them really was probably on the power play, and on the 4-on-4 when they got a little more time and space and they made some good plays," Bratt said. "Other than that, I think we played a really solid defensive game against one of the best offensive teams in the League. Obviously, it was really strong of us to come back to get the win in the end after they scored two goals to tie."
NOTES: Wood, who played the first seven seasons of his NHL career with the Devils, received a video tribute and ovation from the crowd in his first game back at Prudential Center with the Avalanche. He had three shots in 14:50. ... Devils forward Tyler Toffoli was a late scratch due to illness. He was replaced by Max Willman, who had one shot in 8:38. ... New Jersey forward Tomas Nosek had three shots in 10:48 in his return after missing 31 games because of a broken foot. ... The Devils earned their 17th comeback win of the season, tying the Dallas Stars for the NHL lead.