Born in Leeds, West Riding of Yorkshire in 1948, Harvey attended Foxwood School (in the Seacroft area of the city), and played for Leeds City Boys before leaving school to work in a Stylo shoe factory. He was signed as a professional footballer by Don Revie in February 1965, after having played as an apprentice for two years.
Harvey was the reserve goalkeeper for Leeds United during the late 1960s and early 1970s, and was on the reserves’ bench when Leeds won the League Cup and the Fairs Cup in 1968, the League championship in 1969, and the Fairs Cup in 1971.
In the 1970 FA Cup Final, an error by first-choice goalkeeper Gary Sprake allowed opponents Chelsea to equalise in the first half. Sprake was blameless for the second equaliser, but Harvey replaced him for the replay, which Leeds lost 2–1.
In the 1971–72 season, Harvey played only eleven matches, but was selected ahead of Sprake for the 1972 FA Cup Final against Arsenal, as Sprake was suffering from a knee injury. Leeds won 1–0 with an Allan Clarke goal, and from that point on Harvey was the first-choice goalkeeper.
Harvey played 63 times in the 1972–73 season, and played in both the FA Cup Final and the European Cup Winners Cup Final – ending up on the losing side both times.
Leeds put together a 29-match unbeaten start to the 1973–74 season, winning the League championship and earning Harvey the title medal to which he had not been entitled five seasons earlier.
At the start of the 1974–75 season, Harvey took the last penalty during the shoot-out at the Charity Shield game against Liverpool but missed, thus allowing Liverpool to carry the trophy home. Later in the season Harvey was injured in a car crash. He missed the rest of the season replaced by his understudy David Stewart. Stewart was praised for his bravery and “inspirational saves” beating Anderlecht and F.C. Barcelona in the 1974–75 European Cup. Leeds were beaten 2–0 by Bayern Munich in the final in Paris.
The rest of the 1970s saw Leeds decline after the ageing team Revie built broke up. Harvey left in 1980.