The following year, Moog was expected to share goaltending duties with Low, but then a 19-year-old Grant Fuhr surprisingly made the team straight out of training camp.

In 1982–83, Oilers general manager and coach Glen Sather decided to go with the young duo of Moog and Fuhr and traded Low.

After demanding a trade, Moog walked out on the Oilers in 1987 to play for Team Canada at the Calgary Winter Olympics.

In both the 1991 and 1992 Stanley Cup playoffs, the Bruins defeated the Canadiens in the second round to make it to the Prince of Wales Conference Finals.

After the rough start, things improved and Moog backstopped his team to the Adams Division title and the second-best record overall, behind Pittsburgh.

During the middle and latter years of his career, Moog served as vice-president of the National Hockey League Players Association.

In the 1996–97 NHL season, Moog helped Dallas to the Central Division title, but they were upset in the first round in seven games by the underdog Edmonton Oilers.

During his time with the Stars, Moog often shared the workload of regular season games with Darcy Wakaluk and later Artūrs Irbe.

With the Stars bringing in Ed Belfour, Moog signed as a free agent with the Montreal Canadiens for the 1997–98 NHL season.

In the summer of 1998, Moog was offered a contract to play for the Vancouver Canucks, but he chose to retire and instead became the team's goaltending coach.

Moog was selected to play in the National Hockey League All-Star Game four times over his career, twice with the Oilers (1985 and 1986), and once each with the Bruins (1991), and Stars (1997).

[7] In 2016, Moog Joined the Portland Winterhawks of the WHL as their goaltending coach and remained in that position for 7 seasons until he announced his retirement on July 19, 2023, at the age of 63 for health reasons.

Moog also has the highest winning percentage of any goaltender not in the Hockey Hall of Fame (.580), tied with Chris Osgood.