Haula gave Vegas its 44th win when he lifted a wrist shot over Buffalo goaltender Robin Lehner's glove side.
David Perron also scored in the opening round shootout for Vegas, which won for the third time beyond overtime.
Jacob Josefson scored in the third round of the shootout for Buffalo. Ryan O'Reilly and Benoit Pouliot fired wide of the net in the shootout.
Marc-Andre Fleury stopped Sam Reinhart and Jason Pominville in the shootout. He made 31 saves in the first 65 minutes to move within one of becoming the 13th goaltender with 400 career wins.
Lehner stopped William Karlsson in the third round of the shootout while Alex Tuch went wide of the net and Jonathan Marchessault's attempt went off the right post.
Lehner made 33 saves for Buffalo, which is 6-12 in games beyond regulation.
Justin Bailey scored early in the third for Buffalo and Deryk Engelland netted the tying goal late in the third.
Buffalo struck 2:47 into the third on a 2-on-1 break.
Shea Theodore's cross-ice pass from the left point was intercepted by Pouliot, who flew down the right wing and to the right side of the net. Bailey trailed the play untouched, received Pouliot's pass, deked Fleury and scored while falling to the ice.
The Golden Knights tied the game with 4:14 left.
Perron made a pass from behind the net as Lehner lost his balance. Engelland fired a wrist shot from about 10 feet out and it banked off Buffalo defenseman Marco Scandella's right hip and toe and into the net.
The game reached a shootout when Lehner stopped Perron's long wrist shot with two seconds left.
--Field Level Media
Vegas | Buffalo | |
Deryk Engelland 1 | Points | Justin Bailey 1 |
Deryk Engelland 1 | Goals | Justin Bailey 1 |
William Karlsson 1 | Assists | Benoit Pouliot 1 |
N/A | Power Play Goals | N/A |
N/A | Short Handed Goals | N/A |
Marc-Andre Fleury .969 | Save Percentage | Robin Lehner .971 |
Marc-Andre Fleury 31 | Saves | Robin Lehner 33 |
Team | Shots | Goals | Power Play | Penalty Kill | Penalty Mins | Face Offs Won |
Vegas | 34 | 2 | 0-2 | 2-2 | 4 | 19 |
Buffalo | 32 | 1 | 0-2 | 2-2 | 4 | 40 |