The Senators had overcome a 2-1 deficit in the third period on Spezza's second and third tallies of the game, but were forced into an overtime when Mike Ribeiro tipped in an Oliver Ekman-Larsson shot on a power play with 3:31 left in regulation.
Craig Anderson stopped 34-of-37 shots for the Sens, who wrapped up a season-opening six-game road trip with a 2-2-2 mark.
Rob Klinkhammer and Radim Vrbata each found the net and Smith finished with 28 saves for Phoenix, which saw its three-game winning streak snapped.
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