The Senators were trailing 2-0 after two periods, but Jason Spezza halved the deficit 1:21 into the final frame and Erik Condra notched the equalizer on a short-handed tally 12 minutes later.
New Jersey's David Clarkson was whistled in overtime for hooking and Alfredsson one-timed an Erik Karlsson pass into the net during the man-advantage, giving the Sens their fourth straight home win over the Devils.
Craig Anderson earned the victory with 27 saves.
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