Robson was originally spotted playing in a Schools' Cup Final by one of the Clarets' north-east scouting team, Charlie Ferguson. Ian Lawson was playing in the same game and both players signed for Burnley on the same day.

A tall and rangy inside-forward, Robson soon impressed at Turf Moor in the junior teams and signed as a professional on his 17^th birthday in 1956. Later that year, he made his first team debut, coming in for Jimmy McIlroy who was on international duty.

The youngster served early notice of his ability and confidence by crashing home a late equaliser against Blackpool at Turf Moor.

After two seasons on the fringe of the first team, Robson was still a teenager when he established himself in the team during the 1958/59 season, with Albert Cheesebrough the man to make way for the young Jimmy.

The international selectors were also impressed by Robson and in May 1959, he was selected for the England Under 23 side against West Germany in Bochum and he scored in a 2-2 draw.

During the Championship season of 1959/60, Robson was into his goalscoring stride and established a partnership with John Connelly and Ray Pointer. Between them, the trio hit 79 of the team's 100 goals in league and cup. Jimmy's best day undoubtedly came against Nottingham Forest, when he hit five goals in an 8-0 win at Turf Moor.

1960/61 was his best season as a goalscorer with 25 strikes in the League and 12 more in the cup competitions, including the first ever goal against continental opposition in the European Cup, scoring against Rheims at Turf Moor in 1960.

With the emergence of Andy Lochhead and the departure of Jimmy McIlroy, Robson left for pastures new in 1965, signing for Blackpool for £10,000. He spent almost three years with the Seasiders before representing Barnsley and Bury.