The monarch presented the Argentine-born pope with a hamper, containing 18 items from Buckingham Palace, Windsor, Sandringham and Balmoral.
It had two types of honey, a bottle of whiskey, 'Coronation Best Bitter', 'Grandad's chutney', eggs and 'Sandringham handmade aromatherapy soaps'.
In return the Pope presented Her Majesty with a lapis lazuli orb - a semi-precious stone - decorated with a silver cross of Edward the Confessor as a gift for the eight-month-old prince, with an inscription on the base that read ‘Pope Francis, to His Royal Highness Prince George of Cambridge’.
The Queen said Prince George "will be thrilled by that".
The royal couple was also presented with a copy of the 1679 decree by Pope Innocent XI which turned Edward the Confessor into a saint for the Catholic Church.
It was the Queen's seventh meeting with a head of the Roman Catholic Church.