Cooper was born on July 22, 1621 in Dorset, England, the eldest son and successor of Sir John Cooper, 1st Baronet, of Rockbourne in Hampshire, and his mother was the former Anne Ashley, daughter and sole heiress of Sir Anthony Ashley, 1st Baronet. He was at the home of his maternal grandfather Sir Anthony Ashley in Wimborne St Giles, Dorset. He was named Anthony Ashley Cooper because of a promise the couple had made to Sir Anthony.
Cooper's mother died in 1628. In 1629, his father remarried, this time to the widowed Mary Moryson, one of the daughters of wealthy London textile merchant Baptist Hicks and co-heir of his fortune. Through his stepmother, Cooper thus gained an important political connection in the form of her grandson, the future 1st Earl of Essex. Cooper's father died in 1630, leaving Cooper a wealthy orphan. Upon his father's death, he inherited his father's baronetcy and was now Sir Anthony Ashley Cooper.