After completing his secondary education at St Mirin’s Academy in Paisley, Hay signed for Celtic in 1966. He became one of the “Quality Street Gang” reserve team which eventually took the places of the ageing Lisbon Lions. He made his league debut on 6 March 1968 against Aberdeen going on to make 230 appearances for Celtic scoring 12 goals, winning 5 League Championships, 3 Scottish Cups and 1 League Cup. After playing for Scotland in the 1974 World Cup in West Germany, and following a dispute with Celtic, he was transferred to Chelsea for a club record £225,000. Initially signed as a direct replacement for Alan Hudson in midfield, he adopted a more defensive role in the young Chelsea side that emerged following their relegation a year later. In 1979 a serious knee injury forced him to retire from the game as a player. He had also suffered problems with a detached retina and eventually lost full vision in his right eye. He won a total 27 caps for Scotland.

His first venture into football management was when he took over the reins at Motherwell in 1981 following Ally MacLeod. He led Well to the Scottish First Division title and promotion to the Scottish Premier Division. He left at the end of the 1981-82 season.

Hay succeeded Billy McNeill as Celtic manager in 1983, winning the Scottish Cup in 1985 and the Premier Division in dramatic fashion in 1986. Celtic won their final game 5–0 against St Mirren at Love Street but needed Hearts, who were league leaders and favourites to lift the trophy, to lose to Dundee at Dens Park. Albert Kidd secured the victory for Dundee with two goals and Celtic were crowned champions. After Celtic failed to win a trophy in 1986–87, Hay left Celtic. He was asked to resign, but refused, so he was sacked and was replaced by McNeill.

In 1989, Hay ventured to Norway and led Lillestrom SK to the Norwegian Premier League championship.

In 1991, Hay took on Paisley club St Mirren but left a year later.

Hay became chief scout at Celtic in 1994, and was responsible for identifying players such as Jorge Cadete, Paolo di Canio and Pierre van Hooijdonk who signed for the club. He became Assistant General manager to Jock Brown in 1997, but left in 1998 in acrimonious circumstances.

Hay teamed up with Jim Leishman in 2000 as co-managers of Livingston, and led the club to the Scottish First Division title at the end of 2000-01, and the club’s first promotion to the Scottish Premier League. Their debut season in the top flight in 2001-02 earned them their first qualification into Europe with a UEFA Cup place. The 2002-03 season was less impressive and they both stepped down from the management position, although they both stayed at the club. Brazilian Marcio Maximo Barcellos took over.

Hay rejoined Livingston on 16 June 2008 in an advisory role. He briefly became interim manager in 2009 after Paul Hegarty was suspended by the club.

In 2015 Hay returned to Celtic as a director of the Celtic Pools, which raises money for the youth academy. He has also worked in a business development role for New College Lanarkshire.