Robb, who was drafted by the Los Angeles Kings 107th overall back in 1986, turned professional in 1989. He would appear in 2 games with the Kings in 1989-90. Otherwise Robb was buried in the minor leagues until 1992-93.
In that season Robb emerged as an NHL story. The rookie went undefeated in his first 10 starts that season (9-0-1), including a 7 game consecutive winning streak. Robb, who was Kelly Hrudey's back up that season, ended with a 15-8-4 record, and posted another 3 big wins in 4 playoff games as he helped Wayne Gretzky and the Kings go all the way to the Stanley Cup finals. Robb calls that moment his greatest in his hockey career.
Robb's fortunes went downhill quickly the following season, as did the Kings'. Robb struggled through a 4-11-5 season. He did pick up his first and only NHL shutout in a rare 0-0 tie against Dallas.
The Buffalo Sabres were hoping to resurrect Robb's career when they acquired him during the lockout shortened season of 1994-95. Robb was involved in the huge trade which saw Robb, Alexei Zhitnik, Charlie Huddy and a draft pick come to Buffalo in exchange for Phillippe Boucher, Denis Tsygurov and Grant Fuhr. Robb played in 6 games for the Sabres in the 48 game condensed schedule. However because of the emergence of Dominik Hasek, Robb rarely got a chance to play.
That proved to be Robb's final season in the NHL. He played the 1995-96 season with the Sabres farm team in Rochester, where the highlight of his season was when he scored a goal on October 9, 1995. He would sign with the Washington Capitals and New York Rangers over the following 2 seasons, but spent the entire seasons in the minor leagues. He rounded out his career with a short stint with the independant Manitoba Moose.
Upon retirement, Robb returned to Minnesota where he is a goaltending consultant at his old stomping grounds at the University of Minnesota. He also invented the Staubar Trainer which is a device goalies wear in practice which restricts the goalies ability to rely on reflexes or athleticism with the idea being forces the goalie to learn the fundamentals of playing angles and using the bulk of his body to get in the way of the puck.