Harris was a member of the Chelsea side which won the FA Youth Cup in 1961 and made his senior club debut in February 1962 in a 1–0 win against Sheffield Wednesday.
Chelsea, led by Harris, reached another FA Cup final three years later, this time against Leeds United – a side then at their peak in English football.
With Leeds having taken a 2–1 lead at Wembley with just six minutes remaining, it was his quick free kick which led to Ian Hutchinson's headed equaliser to take the game to a replay.
While many of Chelsea's star players departed during the 1970s (due to Peter Osgood and Alan Hudson falling out with manager Dave Sexton, and due to the financial crisis caused by the building of the new East Stand, and the financial impact of relegation), Harris remained ever-present in the side throughout a decade which saw them relegated twice and promoted once, although he was replaced as Club Captain by John Hollins at the tail end of the 1971/72 season.
In the closing stages of his career, he also played as a holding midfielder in a role similar to Claude Makélélé, providing a screen for a defence that was often caught out by counter-attacks.