Parise and Elias both faked shots against Sabres goaltender Ryan Miller before sending the puck past him. Parise scored on the forehand, while Elias slid a backhand shot between the netminder's pads to win it for the Devils.
Paul Martin and Dave Clarkson scored in regulation.
Martin Brodeur, perfect in the shootout, had 21 saves as the Devils won their third straight. New Jersey is 19-5-1 in its last 25 games.
Dan Paille had both goals for Buffalo, which played without injured regulars Maxim Afinogenov (groin), Derek Roy (shoulder) and Tim Connolly (groin). Defenseman Jaroslav Spacek was also sidelined due to back spasms.
Miller had 23 saves.
Celebrity Couples of 2014 [PHOTOS]