Hoffman scored two goals to help the Senators beat the Colorado Avalanche 4-3 on Saturday for a sweep of their two regular-season games in the Swedish capital.
"It's huge," Hoffman said. "Both the games were close, tied till the end. Leaving Sweden with four points definitely feels good."
Oduya, a Stockholm native, drove a slap shot by Jonathan Bernier at 7:01 of the third, tying it at 3 and delighting the crowd of more than 13,000. Hoffman got the game-winning goal on a one-timer on a power play with 6:37 left.
"We were playing a real solid game," Ottawa coach Guy Boucher said. "But what was happening, there was lot of weirdness in the game, deflections ... and we were missing a lot of quality, high-end quality chances."
"For some reason, we get hesitant with the puck in the third period," Colorado captain Gabriel Landeskog, another Swede, said. "That's been the story all season long."
The two games in Stockholm were part of the 2017 NHL Global Series. It was the fifth time the NHL has played regular-season games in Europe, but the first since 2011.
The Senators beat the Avalanche 4-3 in overtime in the opener on Friday.
Ottawa tied it on a fluke power-play goal just 12 seconds into the second. Hoffman chipped the puck toward the net from just past the blue line and it bounced past Bernier for his fifth of the season.
It was just the beginning of a wild period.
Ottawa's Chris Wideman had a goal disallowed at 4:28 due to the goalie interference.
Alexander Kerfoot got the lead back for Colorado when he wristed a power-play shot past Mike Condon at 8:34. Just 42 seconds later, Stone tied it again with his 12th goal, converting a backhander after a perfect pass from Karlsson.
Andrighetto then made it 3-2 Avalanche with his fifth of the season at 10:36, prompting Ottawa to replace Condon with Craig Anderson after the goalie allowed three goals in 10 shots.
Ottawa F Matt Duchene, who came over from Colorado in a three-team deal last Sunday, was held off the scoresheet for the second straight game with his new team. In a blow for Duchene's line, left winger Zack Smith dislocated his left thumb on Friday and was replaced by Alexandre Burrows.
Avalanche: Host the Washington Capitals on Thursday night.