Fisher drove a slap shot from the slot past Cam Ward to restore Ottawa's one-goal lead after Carolina scored goals 1:39 apart earlier in the period to tie it at 2.
Milan Michalek also scored for the Senators, 1-2-1 in their first four games.
Brian Elliott made 19 saves after taking over early in the first for injured Ottawa starter Pascal Leclaire, who left with a lower-body injury.
Carolina captain Eric Staal assisted on Sergei Samsonov's goal 7:20 into the third before drawing the Hurricanes even with his first of the season at 8:59.
Cam Ward stopped 36 shots for Carolina, which made its North American debut six days after completing a sweep of its season-opening, two-game series against Minnesota in Finland.
Leclaire had made three saves when he left 2:06 in after a goalmouth collision with Hurricanes forward Tom Kostopoulos.
Fisher opened the scoring when his high shot got behind Ward 17:39 into the first for his first goal of the season.
Michalek made it 2-0 with his first goal 10:23 into the second. Ward had little chance as the Czech forward finished off a nice passing play with linemates Daniel Alfredsson and Jason Spezza.
Samsonov, who made his season debut, drew the Hurricanes within 2-1 when he beat Elliott after taking Staal's set-up pass 21 seconds after Michalek was called for interference.
Staal tied it at 2 when he steered Erik Cole's pass beyond Elliott at 8:59.
NOTES: Samsonov sustained a neck injury in a preseason game against Atlanta on Sept. 25 and did not accompany Carolina on its season-opening trip to Russia and Finland. ... Alfredsson's assist was his 994th career point.