By the time he was 15, he had yet to be claimed in the Midget Draft so his father drove him to Sault Ste. Marie, Ontario, to tryout with the Greyhounds of the Ontario Hockey League. Vanbiesbrouck made the team, and for three years he was the number one goalie in the Soo. By the time he was eligible for the NHL Entry Draft, there was no doubt that he would be selected. The Rangers chose him 72nd overall in 1981, and just a few months later he made his professional debut with the Blueshirts as an emergency call-up.
On December 5, 1981, he beat the now defunct Colorado Rockies 2-1, but at 18 years of age, he was returned to the Soo to develop. For two years he played goal for the United States at the World Championships, and in the fall of 1984 he made the Rangers full time. But the New Yorkers were eliminated quickly from the playoffs in the spring of 1985, and Vanbiesbrouck gladly accepted an invitation to represent the U.S. at the World Championship in Prague.
He stayed the Rangers number one goalie until Mike Richter arrived on Broadway and established himself as an equal but younger talent. The two became the best duo in the game, but Beezer was left exposed in the Expansion Draft in 1993 by Vancouver after the Rangers had traded him to the Canucks for future considerations (Doug Lidster). The Panthers selected him first in that roster-building draft, and Vanbiesbrouck almost single-handedly legitimized a talent thin team in Florida with his consistent brilliance.
Mature and on his own again in the Florida net, Vanbiesbrouck established himself as one of the best goalies in the game. He was with the team for five years and led them to an improbable run to the Stanley Cup finals in the spring of 1996 before losing to Colorado in four games. In 1998 he found himself an unrestricted free agent for the first time in his career, and he auctioned his services to the highest bidder. That turned out to be the Philadelphia, a Stanley Cup-contending team that had lost faith in its starter, Ron Hextall. Beezer signed a three-year contract for $11.25 million, but his first season with the new Flyers was disappointing. He failed to shine in the regular season, and in the playoffs the Flyers were eliminated in the first round by the Toronto Maple Leafs.
His last full season was 2000-01, which was split between the New York Islanders and New Jersey Devils. After a brief retirement, Vanbiesbrouck played five more games with the Devils before calling it a career on May 22, 2002. He was the 15th goalie, and only the second American-born goalie, a month after Tom Barrasso, to win 300 NHL games.
Internationally, he played in both the 1987 and 1991 Canada Cups for the U.S., though he missed the inaugural World Cup because of shoulder surgery. He teamed again with Mike Richter at the Nagano Olympics in Japan in 1998 when the NHL shut down to allow all its pros to participate. However, the U.S. sixth-place finish was considered a disappointment.
Vanbiesbrouck purchased a share of the OHL's Sault Ste. Marie Greyhounds after retiring and served as the team's GM and coach. However after directing a racial slur towards his captain, Vanbiesbrouck relinguished his duties.