Team owner Terry Pegula said in a statement that the club has already begun the process of filling the positions.
The Sabres missed the playoffs in both seasons under Bylsma, and Murray was never able to guide Buffalo to the postseason in parts of four campaigns as GM.
Bylsma won 68 of 164 games in his two years behind the bench. He was hired by Buffalo in May 2015 after a year away from the NHL following parts of six seasons as head coach of the Pittsburgh Penguins, with whom he won the Stanley Cup in his first year.
Murray was tasked with rebuilding the Sabres when he was hired in January 2014, and he wasn't shy about making significant moves.
At the 2015 draft, he acquired goaltender Robin Lehner and veteran forward David Legwand from the Ottawa Senators for the 21st pick, which the Senators used to select highly touted prospect Colin White.
Thursday's firings came about 24 hours after WGR 550's Paul Hamilton reported Eichel had no desire to sign a contract extension with Buffalo this summer if Bylsma remained head coach.