Brennan was a tough centre half who moved to Newcastle United from Airdrieonians for £7,500 in 1946. He made his debut for the club against Millwall in August 1946 and appeared on 351 occasions for the Magpies between 1946 and 1956.
Brennan moved into management after his playing retirement, initially with North Shields between 1956 and 1962, then in Singapore and Trinidad. He returned to North Shields in 1967 and helped them to victory in the 1969 FA Amateur Cup. He was appointed manager of Darlington in August 1971 but stayed in the post for only three months before joining South Shields in a coaching role in 1972.