The Blue Jackets had forged a 2-2 tie on James Wisniewski's power-play goal 7:16 into the third period, but the Devils went on a power play with less than 7 minutes left when Nathan Horton was whistled for a penalty.
Just before the man-advantage expired, Spezza joined a crowd in front of the net a sent the puck past Curtis McElhinney for the decisive score.
Stephane Da Costa netted a pair of goals for Sens, who completed a 2-1-1 road trip.
Craig Anderson earned the win with 34 saves.
Cam Atkinson and Wisniewski each had a goal and an assist as Columbus dropped a third straight following a club-record eight-game winning streak.
McElhinney ended with 25 stops.