Stevenson began his professional career in 1960 with Motherwell and played in twelve league matches over a two-year period. A 1962 move to Hibernian saw him feature twenty times but leave within a year for Morton. It was with Morton that Stevenson would spend half his career and make the majority of his career appearances, playing in over 160 league matches and scoring twenty-six goals. After six years at Cappielow, Stevenson moved south to Luton Town but played just once and quickly returned north, this time with Dundee United. After three years at Tannadice, Stevenson finished his career with a handful of appearances for Berwick Rangers.