Roberts was selected by the Calgary Flames in the first round (No. 12) of the 1984 NHL Draft after the forward from North York, Ontario, scored 24 goals in 57 games in his second season with Ottawa of the Ontario Hockey League. He played most of his next two seasons in the OHL before turning pro and making his NHL debut with the Flames on Nov. 11, 1986, when he scored a goal against the Vancouver Canucks.
Roberts became a full-time NHL player in 1987-88 and scored 22 goals in 1988-89, when the Flames won the Stanley Cup for the first time. In 1989-90, he scored 39 goals and 72 points, and two seasons later set NHL career highs in goals (53) and points (90).