Moog was one of the most successful goalies of the 1980s and '90s, winning the Stanley Cup three times with the Edmonton Oilers and 372 regular-season games during 18 NHL seasons.

The Oilers selected Moog in the seventh round (No. 132) of the 1980 NHL Draft. He went 3-3-0 with a 3.82 goals-against average in seven games as a rookie in 1980-81 but took the starting job in the playoffs and keyed Edmonton's first-round upset of the Montreal Canadiens.

Moog and Grant Fuhr began sharing the goaltending with the Oilers in 1982-83. He was named the starter in the 1983 playoffs and went 11-5, helping the Oilers reach the Cup Final, where they lost in four games to the New York Islanders. Fuhr was the starter in the 1984 playoffs, but when he was injured in Game 3 of the Cup Final against the Islanders, Moog stepped in and helped the Oilers win their first championship, starting the final two games.

Though Moog and Fuhr split the starting job during the next three seasons, the net was Fuhr's during the playoffs; Moog played in five postseason games from 1985-87, winning the Stanley Cup two more times. Moog left the Oilers and played for Canada's national team in 1987-88, including the 1988 Calgary Olympics, when he won all four of his starts. After the Olympics, the Oilers traded him to the Boston Bruins.

Moog went 4-2-0 in six starts with Boston, then backed up Reggie Lemelin for most of the Stanley Cup Playoffs. The Bruins advanced to the Cup Final, and Moog was in goal for the final game of the series when the Oilers wrapped up their fourth championship in five seasons.

Two years later, Moog helped the Bruins win the Jennings Trophy for allowing the fewest goals during the regular season. Moog was Boston's starter in the playoffs and helped the Bruins return to the Cup Final. He led all goalies with a 2.21 goals-against average, but the Bruins lost to the Oilers in five games.

Moog was Boston's starter for the next three seasons, winning 90 games. But after the Bruins were swept by the Buffalo Sabres in the playoffs, with Moog taking three of the four losses, he was traded to the Dallas Stars on June 25, 1993. During his first season with the Stars, Moog won his 300th NHL game, 6-2 against the Washington Capitals on March 19, 1994. Only Jacques Plante (526 games) had reached 300 wins in fewer games than Moog (543).

The best of his four seasons with the Stars came in 1996-97, when he won 28 games, had a 2.15 GAA and helped Dallas finish first in the Central Division. However, the Oilers upset the Stars in the first round of the playoffs, and after Dallas signed Ed Belfour as a free agent, Moog signed with the Montreal Canadiens on July 17, 1997.

Though Moog helped Montreal win a playoff series for the first time since 1993, he retired after the 1997-98 season. Moog finished 372-209 with 88 ties, a 3.14 GAA, .891 save percentage and 28 shutouts. He was 68-57 with a 3.04 GAA, .890 save percentage and four shutouts in the Stanley Cup Playoffs.