[citation needed] Billington joined the Belleville Bulls of the Ontario Hockey League for the 1983–84 season.

In one post-season game, Billington had a GAA of 6.00 as the Bulls lost to the Oshawa Generals in the Leyden Division quarter-finals.

In the playoffs, Billington posted a GAA of 2.71 in 14 games, as the Bulls lost to the Peterborough Petes in the Leyden Division finals.

Following the season, Billington was named the winner of the Bobby Smith Trophy, awarded to the OHL Scholastic Player of the Year.

In 1985–86, Billington returned to the Bulls late in the season after spending most of the year with the New Jersey Devils of the NHL.

The Devils assigned Billington to his junior club, the Belleville Bulls of the Ontario Hockey League for the 1984–85 season.

Late in the season, Billington was assigned to his junior club in Belleville in an effort to get him more playing time.

Billington split the 1986–87 season between the Devils and their American Hockey League affiliate, the Maine Mariners.

On December 28, 1991, Billington stopped all 26 shots he faced in a 3–0 win over the Buffalo Sabres for his first career shutout.

On April 20, 1993, Billington played in his first career NHL playoff game, as he made three saves on four shots in relief of Terreri in a 7–0 loss to the Pittsburgh Penguins.

On April 22, Billington made his first career NHL playoff start, as he stopped 31 shots in a 4–3 loss to the Penguins.

After starting the season 0–4–1, Billington earned his first victory with Ottawa on October 25, as he made 23 saves in a 4–1 win over the Mighty Ducks of Anaheim.

On April 9, 1995, Billington made his Bruins debut, making 22 saves in a 6–5 win over the Buffalo Sabres to earn his first victory of the season, after going 0–6–2 with Ottawa.

In his next start, on October 28, Billington stopped all 40 shots he faced in a 1–0 victory over the Washington Capitals for his first win with the Avalanche.

Billington made his debut with the Washington Capitals on October 19, 1999, as he played the third period against the Mighty Ducks of Anaheim in relief of starting goalie Olaf Kolzig.

On October 31, Billington made his first start with the Capitals, making 20 saves in a 2–1 loss to the San Jose Sharks.

In his next start, on November 20, Billington stopped all 28 shots he faced in a 3–0 victory over the Boston Bruins for his first win as a member of the Capitals.

Upon his retirement, Billington joined the Colorado Avalanche's staff,[2] eventually working up to an assistant general manager position in 2009.

In the 2016–17 season, he was named the general manager of the Avalanche's American Hockey League affiliate, the San Antonio Rampage.