Stevenson made his debut in the inaugural Football League season on 8 September 1888, at Anfield, then home of Everton.
He played at full–back in a defensive line that kept two clean-sheets and limited the opposition to one goal on five separate occasions.
On 20 October 1888, Stevenson was part of a defence that prevented 'The Invincibles' of Preston North End from scoring; Preston played 27 League and FA Cup matches in that season and their visit to Accrington was the only time they failed to score a goal.
The match had to be replayed as West Bromwich Albion protested at the state of the Thorneyholme Road pitch.
[3] In the earlier part of his career with Arthurlie, he was selected for a trial match for the Scotland national team in 1881, though no further call-ups followed.