In the 1977–78 season, at the age of eighteen, Briggs signed for Middlesbrough but didn't make any first-team appearances for the club. Later that season, he moved to Oxford United. The fee was settled at the Football League's first-ever transfer tribunal. Briggs spent eleven years at the Manor Ground, where he received the nickname "Rambo" and became a cult hero, winning the club's "Player of the Year" accolade three times. He formed a successful central-defensive partnership with club captain Malcolm Shotton as United won three trophies between 1984 and 1986: the Division Three championship in 1983–84, the Division Two championship the following season, and the League Cup in 1986.