Carriera, who was one of the most fashionable artists of her day, created portraits of many notable figures, including Louis XV of France, English author Horace Walpole and French painter Antoine Watteau.
Her works are in the Louvre in Paris, The National Gallery in London, The Frick Collection in New York and other major galleries.
The previous sale record for a Carriera painting was £421,250, for a portrait of Irish soldier and politician Gustavus Hamilton sold in New York in 2002, Cheffins said.
The auctioneer’s Luke Bodalbhai said the Fellowes painting exhibited Carriera’s “talent as one of the leading lights of 18th Century portraiture”.