Coton started his professional career at Birmingham City in 1978, having signed from Mile Oak Rovers the previous year. He made his Football League debut as a 19-year-old, on 27 December 1980 in the First Division match against Sunderland. His first touch of the ball was to save a penalty awarded after 54 seconds. By the 1982–83 season he had established himself as Birmingham's first-choice goalkeeper and was named Player of the Year.
Watford
Birmingham were relegated from the First Division at the end of the next season, but Coton found his way back into the top flight with a transfer to Watford, for a sum of £300,000. He soon replaced Steve Sherwood in Watford's goal and at the end of his second full season at Vicarage Road he won the Hornets' Player of the Season and Display of the Season awards, the latter for a clean sheet against Liverpool. He remained with the club even after their relegation from the First Division in 1988. He went on to become Watford Player of the Season for an unprecedented third time in 1989–90. Coton later became the second player to be inducted into Watford's Hall of Fame, behind club legend Luther Blissett.
Manchester City
Before the start of the 1990–91 season, Coton was bought for just under £1 million by Manchester City managerHoward Kendall. At the time, he was one of the most expensive goalkeepers to be signed by any British club. He went on to win the City Player of the Year award in both the 1991–92 and 1993–94 seasons. They finished fifth in the First Division during his first two seasons at Maine Road and ninth in the first season Premier League. However, they dipped to 16th in 1993–94 and 17th in 1994–95, although he remained first-choice goalkeeper under subsequent managers Peter Reid and Brian Horton.
Coton's time at Manchester City was cut short by injury in 1995, and the signing of Eike Immel. In January 1996, he moved across the city to Manchester United for £500,000, a record fee for a transfer between United and City, to be understudy to Peter Schmeichel. He never played a first team game for United, made the substitutes' bench only once, and moved on to Sunderland after just six months, for a fee of £600,000.
Sunderland
He made a few impressive performances for his new club before breaking his leg in five places during a league match against Southampton. The injury marked the end of his playing career.
Hereford United
In April 2004, while a coach with former club Manchester United and in his 43rd year, he was brought in as emergency goalkeeping cover by Hereford United for a Conference fixture.
International
Coton won an England Bcap during his time at City, and was also called up to the senior team by Graham Taylor in November 1993.
Later career
From 1997 to 2007, Coton was goalkeeping coach at Manchester United. He was forced to step down from the position in December 2007 due to a knee injury that stopped him participating in training sessions. He had had two operations in four months to correct his ongoing knee problems, but after discussions with his surgeon and the Manchester United medical staff, they agreed that retirement was the best option. Coton remained at Manchester United until the end of his contract in June 2008, but with reduced coaching responsibilities. He went on to become a player's agent. Coton spent time as chief scout of Wigan Athletic in 2014, and also scouted for Bolton Wanderers, before joining Aston Villa in 2015 as goalkeeping scout. He then spent two years as their head of domestic scouting, before returning to Sunderland as head of recruitment in June 2018.