Leighton joined Aberdeen from Scottish Junior club Dalry Thistle in 1977,[5] after working in the civil service on leaving school.

Leighton played when United were defeated 5–1 in the league by newly promoted neighbours Manchester City, and conceded three goals in 1990 FA Cup Final, a 3–3 draw with Crystal Palace.

Although Leighton remained contracted to United for nearly two more years, he played only one more game for the club (against Halifax Town in the Football League Cup on 26 September 1990).

He was fourth choice keeper in 1990–91 behind Sealey, Gary Walsh and youngster Mark Bosnich, and was loaned to Arsenal between March and May 1991, a period in which they won the league title, although he did not play.

[3] Leighton remained fourth choice United keeper in season 1991–92 behind Peter Schmeichel, Walsh and youngster Ian Wilkinson.

[11] He played in the last dozen league games of the season winning the title in the Scottish second tier and promotion to the top flight.

[3] His career was revitalised after Alex Miller signed him for Hibernian in summer 1993,[3] Leighton playing over 150 games and missing only one league match in four seasons.

In that game he was substituted after three minutes, suffering a broken jaw in a challenge from Rod Wallace (requiring the insertion of metal plates into his face).

[15] In July 2000 he was given a testimonial against Middlesbrough,[16] the same opponent as in his first appearance for Aberdeen in a 1978 friendly;[17] however he could not actually play in the match due to recovering from his cup final injury.

[3] Leighton regained the position after a man of the match performance in a 1-0 home win versus Sweden in November 1996[3][18] during qualification for the 1998 World Cup.

This set a new record as the oldest player to play for Scotland, since broken by David Weir in September 2010[19] and later by Craig Gordon in June 2024.

He parted company with the club in August 2009 when manager Mark McGhee (his former teammate, who had also brought him to Reading years earlier) chose to replace him with Colin Meldrum.