With 3:47 elapsed in the extra time, Psysk's shot the right side was initially stopped by Smith, who deflected it straight up in the air.
It came to rest in the back of his pants, but Smith, unaware of where the puck was, slid backwards into his own net for the game-winning goal.
Tyler Ennis had forced overtime with an equalizer 14:01 into the third period for the Sabres, who notched a third win in four games.
Ryan Miller recorded the victory with 36 saves.
Martin Hanzal scored the sole goal for Phoenix, losers of four in five games.
Smith finished with 30 saves.
Notable deaths of 2014 [PHOTOS]