The Lightning were trailing 2-1 early in the third period when Teddy Purcell knocked in a shot from the slot during a power play for the equalizer.
The contest remained knotted until the overtime, when Killorn got control of the puck in front of the net, skated to the right side and lifted a backhand shot past Jhonas Enroth for the game-winner.
Tyler Johnson also scored and Ben Bishop stopped 21-of-23 shots, including all 13 he faced in the third period, as the Bolts won for the second time in three games starting the season.
Enroth finished with 31 stops for the struggling Sabres, who fell to 0-3-1.
Cory Hodgson and Jamie McBain each tallied for Buffalo.