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It was during his initial spell at Aberdeen where Leighton achieved his greatest success, under the management of Alex Ferguson in the early to mid 1980s. Ferguson's Aberdeen side won the Scottish Cup four times, the Scottish League Cup twice, as well as the Cup Winners' Cup in 1983 and two Scottish Premier League titles in 1984 and 1985.

In 1988, Leighton followed Ferguson to Manchester United. His first season at Old Trafford was relatively successful but his second was marked by a loss of form and confidence. His move to Hibernian marked a revival in his fortunes, including a return to the national team. Leighton returned to Aberdeen, retiring from club football after the 2000 Scottish Cup final, which unfortunately saw him taken off injured after just 3 minutes. With forward Robbie Winters taking over goalkeeping duties, Aberdeen lost 4-0 to Rangers.

Leighton's international career stretched over 16 years, from 1982 to 1998. He was an uncapped squad member for the 1982 World Cup, before making his debut against East Germany on October 13, 1982. Within two years he was his country's No. 1, and he played in all of his country's matches at the 1986, 1990 and 1998 World Cups, despite stiff competition for the keeper's jersey from Andy Goram during the 1990s. He retired from international football in October 1998, at the age of 40. Leighton is a member of the Scotland Football Hall of Fame, having won his 50th cap for Scotland in 1989.