Although born in the town of Easington, County Durham, Brennan played for the Chesterfield and Derbyshire schoolboys' teams before joining Burnley as an apprentice following a successful trial.
Brennan eventually made his senior debut for Burnley on 6 April 1974, deputising for the unavailable Keith Newton in the 0–4 defeat away at Stoke City.
Brennan scored his first goal for Burnley in the 1–0 win against Liverpool on 24 September 1974, beating England international goalkeeper Ray Clemence with a well-struck shot from 30 yards out.
Brennan was almost ever present over the next two-and-a-half seasons and was part of the team that won the Anglo-Scottish Cup in 1978, beating Oldham Athletic over two legs in the final.
Brennan retired from professional football in the summer of 1982 and subsequently moved into non-league, assisting Burnley Belvedere and later Colne Dynamoes.