[2] Born in Belfast, Brennan began his career in his home country of Northern Ireland, firstly with Harland and Wolff Welders and then Bloomfield United and Distillery.

He left the club to play for Great Yarmouth Town before returning to Norwich in 1957.

In the 1958–59 season, Brennan was a member of the Norwich team that reached the semi-finals of the FA Cup as a Third Division side.

It was one of the most famous FA Cup runs in the history of the competition, as the club beat First Division teams Manchester United and Tottenham Hotspur along the way.

In the semi-final against his former club Luton, Brennan scored Norwich's equaliser in the first match at White Hart Lane before the team lost the replay at St Andrew's, Birmingham.