Norris pushed in a rebound with 5:54 left and the Senators hung on for their league-low fifth victory of the season.
Anton Forsberg made 47 saves for Ottawa in his seventh game with the Senators and 63rd of his career. He previously played for Columbus, Chicago and Carolina.
The Hurricanes dropped their season-high third game in a row despite holding a 49-20 edge in shots.
Teuvo Teravainen scored with 7:58 left on what was the 42nd shot of the game for the Hurricanes. Andrei Svechnikov tied it with 6:13 to go.
Antti Raanta, who suffered his first regulation loss of the season in five decisions (3-1-1), made 17 saves for Carolina.
Tyler Ennis notched two assists for the Senators, who had allowed at least four goals in each of their six previous games.
Ottawa went up 2-0 with 10:55 remaining in the third period. The puck went into the net off Carolina forward Martin Necas’ skate, but goaltender interference had been ruled on Formenton.
The Senators challenged the ruling, and it was overturned. That gave Formenton his second goal of the game and fourth of the season.
Carolina held a 20-3 edge in shots through the first period, but Ottawa owned the lead. Formenton’s first goal came when the Senators completed a nifty transition sequence.
The Hurricanes failed to score a first- or second-period goal for their third consecutive game. Svechnikov’s shot bounced off the post on a power play in the waning seconds of the first period.
The Hurricanes played their second game in a row without defensemen Tony DeAngelo and Brett Pesce. They entered the COVID-19 protocol earlier in the week before the team’s trip to Dallas.
Defenseman Ethan Bear, who entered the COVID-19 protocol last week, is back with the team but did not play Thursday night.
Center Adam Gaudette, who scored a goal in his Senators debut Wednesday night, was a scratch for Ottawa.
Senators: Host Colorado on Saturday night.
Hurricanes: Host Buffalo on Saturday night before a five-game trip.