Hasek gave New York fits, going unbeaten in 14 of his last 16 appearances against the Rangers. Since his departure from Buffalo, the Sabres were 0-2-1-1 against their in-state rivals.
But Biron blanked New York after giving up a power-play goal to Eric Lindros with three minutes remaining in the first period.
While Bobby Holik is back in the lineup after missing six weeks, the injuries are starting to mount for the Rangers. Defenseman Brian Leetch missed his second straight game with a bruised left ankle and Matthew Barnaby was scratched with a mild MCL strain. Pavel Bure left in the second period with a sprained right knee and will have an MRI on Saturday.
Lindros' goal appeared to give the Rangers momentum, but Holik and Ronald Petrovicky took penalties before the period was over, and Satan cashed in during the five on three to get the Sabres even with 21 seconds remaining.
Buffalo took control in the second, grabbing a 3-1 lead on goals by Satan and Dmitri Kalinin 58 seconds apart midway through the period.
Satan completed his hat trick with 1:29 remaining in the contest, scoring into an empty net.
The Sabres, who entered the contest with a league-low 15 points, posted just their second road win of the season (2-9-2).
Dan Blackburn turned aside 22 shots for New York, which missed another chance to move over .500 after dropping a 3-2 overtime decision in Philadelphia on Thursday.