Nico Hischier scored for the Devils (3-1-0), who played their fourth straight home game to start the season.
Dylan Cozens scored his first goal of the season, and Dustin Tokarski made 37 saves for the Sabres (3-1-1) in their first road game. Buffalo lost 4-1 to the Boston Bruins on Friday.
"The back half of the game, [the Devils] got some momentum," Tokarski said. "We were a little tired, but the boys played a great game. Their goalie played awesome. We could've had a few more. Give him lots of credit. In the overtime, I don't know exactly what led to the play, but he made a shot."
Daws, a third-round pick (No. 84) by the Devils in the 2020 NHL Draft, is the fifth-youngest goalie in their history to make his NHL debut (20 years, 305 days). Craig Billington, who made 23 saves in a 5-2 win against the Los Angeles Kings on Oct. 26, 1985, is the youngest (19 years, 45 days).
"Obviously, I had nerves in the first couple minutes of the game, the legs are a little shaky, but that's expected in my first NHL game," Daws said. "It's a big crowd and I'm not really used to that, but after a few shots, a couple saves, I think I calmed down."
Cozens gave the Sabres a 1-0 lead on the power play with a wrist shot from the top of the right circle at 3:19 of the first period.
"I thought our guys battled really hard," Buffalo coach Don Granato said. "We didn't give up a lot of second chances, we cleaned it up very well. We kept a lot of shots to the outside."