Peters signed his first professional contract in November 1960. His first manager Ted Fenton left the club in 1961, to be replaced by Ron Greenwood. He was a major influence on Peters and his progress as a young footballer. In his first years with West Ham, Peters had played in both defensive and midfield positions before Fenton had encouraged him to play as a right-half. He made his debut on Good Friday 1962 in a 4–1 home win against Cardiff City. He scored his first goal for West Ham in a 6–1 win at Manchester City on 8 September 1962. Also in 1962, Peters played in goal for West Ham in a game against Cardiff City after regular goalkeeper, Brian Rhodes had been injured.