A native of Red Deer, Alberta, he was the Nordiques first pick, 11th overall, in the 1981 Entry Draft. Moller starred with the Lethbridge Broncos of the OHL and was named a second team All-Star in 1981-82, his final year of junior. That year he scored 20 goals and 75 points in 60 games. He also picked up 249 PIMs. He was part of Team Canada at the World Junior Championships, picking up 3 assists in 7 games.
Moller played his first seven NHL seasons with the Nordiques. Though he spoke no french when he arrived in Quebec City, Moller would quickly fall in love with the city, unlike some english speaking players who dreaded life in the predominantly french town.
Though he was an offensive power in junior hockey, Moller used his big 6'2" 210lb body to excel defensively at the NHL level. He lacked speed and a hard shot but Moller was one of the better d-men when it came to making the perfect breakout pass. Many times during his career he would spring lose a Peter Stastny or Mike Gartner with a perfect outlet pass from his own zone. His unfailing accuracy in this regard made him so valuable to his team's transition offense.
Moller enjoyed his finest offensive years while in a Nordiques uniform, registering more than 23 points on four occasions. During the 1985-86 campaign, he set career-highs with seven goals, 22 assists and 29 points and in 1988-89 he matched those totals.
Prior to the 1989-90 season, Moller was traded to the Rangers in exchange for Michel Petit. After two and a half seasons in New York City he was then dealt to the Buffalo Sabres midway through the 1991-92 campaign in exchange for Jay Wells. He played two and a half years with the Sabres before signing with the Florida Panthers as a free agent prior to the 1994-95 campaign.
But after Randy signed with the Panthers, injuries limited his play and ultimately forced his retirement. Back injuries limited him to only 17 games with the Panthers. He accepted the team buyout, but felt bad that he couldn't fulfill his contractual obligations.
Moller retired from the NHL with career totals of 45 goals, 180 assists and 225 points in 815 regular season games.
Like many hockey players before him, Moller turned to the world of broadcasting after hanging up his skates. However he took the unconventional route of becoming the Florida Panthers play by play man. Most players are analysts offering their opinion. Moller, who works alone and provides his own colour commentary, has a very unconventional style.