Kevan was the first signing made by the former Tottenham Hotspur full-back Vic Buckingham after he had taken over from Jesse Carver as manager of West Bromwich Albion in February 1953.
Coached by the club's former striker W. G. Richardson, Kevan had to wait until August 1955 to make his West Bromwich Albion League debut, a 2–0 home win over Everton in which he scored twice after being selected in place of the injured Ronnie Allen.
Kevan scored 20 goals in the 1956–57 season, which included a run to the FA Cup semi-finals where Albion lost to Aston Villa.
[7] He added 18 in 1960–61 and then claimed 33 League goals the following season, when he finished joint top scorer in the First Division with Ipswich Town's Ray Crawford.
His spell at Stamford Bridge under Tommy Docherty was short and unsuccessful, and in the close season of the same year he moved to Manchester City for £35,000 just after their relegation to the Second Division.
At Crystal Palace he made 21 League appearances and scored five goals[13] before moving to Peterborough United in March 1966,[14] and then Luton Town.
[17] In tribute to Kevan, West Bromwich Albion players wore black armbands for their FA Cup match the day after his death.
He scored on his debut against Scotland at Wembley in a 2–1 win in April 1957, and netted twice in the 1958 World Cup Finals in Sweden, against the Soviet Union and Austria,[4] after he was selected for the squad in preference to Brian Clough.