Mittelstadt scored a power-play goal with a tip-in from the slot at 10:20 of the third period to give the Sabres a 4-3 lead. Dahlin kept the puck inside the blue line at the point before passing to Mittelstadt from the right face-off circle.
"We don't win that game without [Dahlin] rising to the level he can rise to," Sabres coach Don Granato said. "He made a couple of big plays to tip that game back our way. That play is exactly why we need him playing with confidence. When you're not confident, you second guess the bounce of that puck, you hesitate, and then it's a break going the other way.
"But when you're in the zone, you react on instincts, and his instincts are incredible."
Buffalo (9-23-6) extended its point streak to five games and is 3-0-1 since an NHL shootout-era record 18 games without a win (0-15-3) ended with a 6-1 victory against the Philadelphia Flyers on March 31.
Linus Ullmark made 27 saves for the Sabres, who host the Devils on Thursday in the second of the home-and-home set.
"Obviously it's been documented pretty well we had a tough stretch there, but it's good to get rewarded," Mittelstadt said. "I think we work hard and it's good to get rewarded. It's good to have the guys have smiles on their faces, see guys having fun. We have to remember the feeling and keep going."
Pavel Zacha and Jesper Bratt each had a goal and an assist, and Scott Wedgewood made 13 saves for the Devils (13-18-6), who are 0-2-2 in their past four games.
"We just didn't play good enough today," Bratt said. "I think we had some good looks but we need to be better as a group. It's a hard division and you've got to compete. I think we have another gear and we need to get it fixed up for next game."
Rasmus Ristolainen scored an empty-net goal at 18:29 for the 5-3 final.
Damon Severson gave the Devils a 1-0 lead off a rebound from the high slot at 4:22 of the first period. Tyce Thompson had the primary assist in his NHL debut.
"It was crazy with emotions, excitement, nerves, but I thought the guys in the room did a good job of keeping me calm," Thompson said. "Just kind of having fun out there. Too bad that we didn't do enough to win."