Norris scored the go-ahead goal with 5:54 left and the Senators hung on for their league-low fifth victory of the season.
Ottawa had lost six games in a row, while also having two other games postponed during that stretch because of COVID-19 protocols.
Anton Forsberg made a career-high 47 saves for Ottawa.
''That's sometimes what you need to get out of something is your goalie to stand in there and give you a gutsy effort,'' Senators coach D.J. Smith said.
Forsberg was in his seventh game with the Senators and 63rd of his career. He previously played for Columbus, Chicago and Carolina.
''If he's not in the net and playing like that, then we probably don't win,'' Senators center Chris Tierney said. ''But he was.''
The Hurricanes dropped their season-high third game in a row despite holding a 49-20 edge in shots.
''We did everything right, but didn't get the win,'' coach Rod Brind'Amour said. ''If we get that kind of effort night-in, night-out, we're going to win a lot of games.''
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.
''When things got tough, we kept pushing through it and found a way,'' Tierney said.
The decisive goal went in off Carolina defenseman Ian Cole after goalie Antti Raanta made an initial stop.
''A couple of bad bounces,'' Raanta said. ''A tough way to lose when you dominate a game. . Nothing really went our way.''
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. A night earlier, they lost 6-2 at home to Vancouver.
''We had to roll four lines with on the back-to-back with travel,'' Smith said. ''There really wasn't a passenger tonight.''
At the end of November, the Hurricanes' 15 wins were tied for the second-most in the league. Now they've dropped three of their last four home 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.
''After that (busy first period), I was just trying to keep rolling with it,'' Forsberg said.
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.
The Senators improved to 2-7-1 in road games, with the other victory coming Oct. 29 at Dallas. Since then, they were 0-6-1 away from home and outscored by a combined 34-17 in those games.
Meanwhile, Carolina has suffered consecutive home losses in regulation. That never happened last season.
''If you don't come to play against these guys, they're going to walk all over you,'' Tierney said.
Senators: Host Colorado on Saturday night.
Hurricanes: Host Buffalo on Saturday night before a five-game trip.
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