 He made his debut in the 2–1 home league defeat to Burnley on 3 November 1962, he scored his first goal in a 5–0 victory over Leyton Orient at Anfield, Stevenson managed to hit the net in the 35th minute, Roger Hunt with a hat-trick and Ian St John scored the others.Stevenson helped the Reds gain the 1963–64 First Division title in only their 2nd season back in the top flight of English football, he featured in 38 of the 42 league games thus cementing a starting role on the left of the Reds midfield.He played a major role in winning of Liverpool's first ever FA Cup in 1965, he played in all of the ties scoring the second goal from the penalty spot in the 2–0 semi-final victory over Chelsea at Villa Park.