Brand signed for Rangers after impressing manager Bill Struth whilst playing in a schoolboy international against England at Wembley in 1952. Struth signed him on a provisional contract in the summer of that year. In 1954 he turned professional and made his debut for Rangers on 6 November against Kilmarnock, scoring two goals in a 6–0 win. He missed the next two seasons doing his national service. After his return in December 1957, he formed an effective strike partnership with Jimmy Millar, and scored 14 goals in 28 games over the course of the second half of the season.
His best season came in 1960–61, where he played in all of Rangers' 34 league games, scoring 24 goals, and 44 in all competitions. These included 5 goals in Rangers' run to the final of the Cup Winners Cup, 3 of which were scored in an 8–0 victory over Borussia Mönchengladbach.
Brand played in four Scottish championship winning sides, 1958–59, 1960–61, 1962–63 and The Treble winning side of 1963–64. He won four League Cup and four Scottish Cup winner's medals, and is the only player to have scored in four consecutive Scottish Cup finals. He played in a total of seven finals for Rangers, scoring six goals and never finishing on the losing side. He played for Scotland eight times, scoring eight goals, although his international opportunities were limited because Denis Law was an established striker in the side.
He played his last match for Rangers on 23 April 1965 when he scored the only goal in a 1–0 win over Third Lanark in the last League game of the season. Brand was sold to Manchester City in August that year for £30,000. Two years later he moved to Sunderland and finished his career at Raith Rovers. He retired in 1970. In his time at Rangers, he played 317 times and scored 206 goals. He is the club's third top post war scorer behind Ally McCoist and Derek Johnstone.
Brand had a brief managerial career, taking charge of Darlington for 6 months from December 1972, then Albion Rovers between 1973 and 1974. After leaving the footballing business he worked as a taxi-driver.