Wayne Gretzky, who owns a share of the team and is its managing partner, will return for a third season as coach next fall.
But the front office felt the fallout from the team's third consecutive last-place finish in the Pacific Division. Also dismissed were director of hockey operations Cliff Fletcher and assistant general manager Laurence Gilman.
The Coyotes finished last in the Western Conference for the first time since the franchise came to Arizona and their 67 points were the team's fewest since that move. Phoenix hasn't made the playoffs since 2002.
Barnett, Gretzky's close friend and his agent for 21 years, had signed a four-year contract extension before last season. He had been general manager since 2001, when Gretzky - considered the greatest player in NHL history - joined the organization as managing partner.
Fletcher, who entered the Hockey Hall of Fame in 2004, was hired as Phoenix general manager in 2001 and promoted to executive vice president of hockey operations the following year. Fletcher was general manager of the Calgary Flames for the first 19 years of the franchise's existence. He oversaw the start of the franchise in Atlanta 1972 and its move to Calgary in 1980.
He was chief operating officer, president and general manager of the Toronto Maple Leafs from 1991-97.
Gilman spent 13 seasons with the Coyotes, the last six as assistant general manager. Earlier, he held several positions, including director of hockey administration and legal counsel.