Stevenson played for Leeds United, Birmingham City and Bristol Rovers.
He started his career at Leeds United in 1972, who were then one of the forces in European club football. Originally thought of as a potential replacement for either Jack Charlton or Norman Hunter as the Don Revie Leeds team started to break up. Stevenson had to wait until April 1976 before he got anything like a run of games when he made 6 successive league appearances replacing Paul Reaney at right back. He had a good run at centre half in 1979 and in midfield in 1981 making a total of 95 appearances.
In 1981 a swap deal which saw him move to Birmingham City where he
played over 100 times. He joined Bristol Rovers in 1985, but decided to retire the following summer aged 29, after struggling with injuries.
Following his retirement from football, Byron became the landlord of the New Inn public house on Elland Road, Churwell, Leeds which also had been managed by another Leeds United and Wales international legend John Charles years earlier.
Stevenson died of throat cancer in September 2007 - one day before what would have been his 51st birthday.