Stanton is the great-great nephew of Michael Whelahan, one of the founders of Hibernian FC and its first captain. He signed for the club in 1963 and made his professional debut later that year. He established himself in the Hibs first team, playing either in defence or midfield. Stanton captained Hibs when they won the 1972 Scottish League Cup Final.
After leaving the club as player and manager, Stanton helps Hibernian with their matchday hospitality. Following complications and surgery after the birth in 2012 of his twin grandsons, Stanton has been devoting his time to support of brain-damaged children.