Spezza first found the net only 55 seconds into the contest, when he cruised in on Cam Ward unimpeded and put a wrist shot past the Carolina goalie.
He struck again on a power play with 2 minutes remaining in the opening period to break a 1-1 tie and put the Senators on the road to only their fourth win in 11 games.
Rookie defenseman David Rundblad notched his first NHL goal and Zack Smith also scored, while Craig Anderson made 34 saves in the victory.
Ward finished with 21 saves.
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