Brady Tkachuk and Connor Brown also scored for Ottawa. Gustavsson was playing in his 12th NHL game.
The 22-year-old Norris, the 19th overall pick in the 2017 NHL draft by the San Jose Sharks, was playing his first NHL game in the U.S. In his third season, Norris played just three games, all in Ottawa, in 2019-20.
The Senators played in the U.S. for the first time since 2019-20, after opening the season with five of their first six games at home and another contest in Toronto.
“We won. That’s the main thing I’ll remember,” Norris said. “I had a lot of fun. … People are standing up when we’re skating around before the game. They were drinking their beer and whatnot. It was electric.”
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Gustavsson noted the crowd at the American Airlines Center.
“This is a huge building with great fans,” he said. “I came out a little bit nervous. You stop the first one and you get on a roll.”
Joe Pavelski scored in the second period for Dallas. Braden Holtby had 20 saves for the Stars.
On the game’s first power play, Tim Stutzle’s shot was blocked and Norris picked up the puck in front and scored at 5:49 of the first period.
Tkachuk scored Ottawa’s second unassisted goal at 7:55 of the first period on a snap shot from the slot for a 2-0 lead.
“Scoring first on the road, and knowing how good this [Dallas] team is in the first,” Senators coach D.J. Smith said, “they can jump all over and you’ve got to be willing to play a north-south game, and I think we really did that in the first, built ourselves a lead.”
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Norris’ second goal came on a pass from Drake Batherson between the circles. Norris scored past Holtby at 7:41 of the second for a 3-0 lead.
Centering a line with Tkachuk and Batherson, Norris moved into the team lead with five goals this season.
“That line continues to be dangerous,” Smith said. “They’re gonna get better. They’re all young guys [Tkachuk, 22, and Batherson, 23] that are learning to play together. They’ve got some physicality. They’ve got a net presence.”
The Stars scored 35 seconds into their first power play, at 10:38 of the second period. Pavelski took a cross-ice pass from Alexander Radulov and sent the puck past Gustavsson.
Dallas had an 11-2 advantage in shots on the goal in the second, but still trailed 3-1.
Early in the second period, Gustavsson turned aside a short-range shot by Jason Robertson. “I saved it with my arm, and then we scored after that.”
Dallas coach Rick Bowness summarized his team’s second-period frustration.
“In the second period, we took it to them,” Bowness said. “They had two shots on net and one goal. We had a ton and had one goal.”
Ottawa added another goal at 5:07 of the third. Brown scored from just to the left of the net on a pass from Victor Mete.