Born in Dublin, Conway began his career with Stella Maris. From there he moved to Bohemians in 1964 as a senior in his home city. In 1966, he moved to Fulham. A midfielder or winger, he spent ten years at Craven Cottage, scoring 67 times in 314 League games before a £30,000 fee brought him north to join Manchester City in August 1976. He was a member of the Fulham side that reached the 1975 FA Cup Final. He played with his brother John at Fulham and his brother Tom also played professionally. Having played just 13 times for City, scoring the winning goal in the final game of the season against Coventry City when Manchester City came second to Liverpool in 1977, he moved to the Portland Timbers of the North American Soccer League for £10,000 on 17 January 1978. He spent three seasons with the Timbers.