In his first professional season, 1986–87, Roberts shuttled between Calgary and Moncton. He was recalled to the Flames three times during the season and scored his first NHL goal in his NHL debut on November 11, 1986, against the Vancouver Canucks. He recorded 15 points in 32 games with Calgary and added 38 points in 38 AHL games with Moncton. In his first full season in Calgary, 1987–88, Roberts improved to 28 points in 74 NHL games, while his 282 penalty minutes were ultimately the highest total of his career, and the first of five consecutive seasons which he recorded over 200 minutes in penalties. Roberts joined the Flames as a grinder; He played a physical style and frequently engaged opponents in fights, but credited Nieuwendyk with helping him establish his place as a power forward and offensive threat with the team. Playing on a line with Nieuwendyk and Håkan Loob, Roberts scored 22 goals in 1988–89. He added 12 points in the 1989 Stanley Cup Playoffs, including two goals in the Flames' 5–3 victory in the fourth game of the Smythe Division final that eliminated the Los Angeles Kings. Roberts and the Flames went on to defeat the Montreal Canadiens in the final to earn the franchise's first Stanley Cup championship.