The Goldfish Pool at Chartwell, which depicts the pond at his Kent home, was painted in 1932.
It was auctioned at Sotheby's in London on Wednesday, following the death of his daughter Mary Soames in May.
The previous auction record for one of his paintings was £1m. Fifteen of his paintings were included in the sale of Mary Soames' possessions, which raised a total of £15.4m.
Sotheby's described The Goldfish Pool at Chartwell as "undoubtedly Churchill's masterpiece from the decade" and "a striking manifestation of the artist at his very best".
It had been given an estimated value of £400,000-£600,000.
Another painting, Tapestries at Blenheim, sold for £1m, while his depiction of The Harbour, Cannes, fetched £722,500.
Among the other possessions, a red ministerial despatch box he used while secretary of state for the colonies from 1921-22, far exceeded its estimated value of £5,000-£7,000, eventually going under the hammer for £158,500.
Churchill was Britain's prime minister during World War Two. He died in 1965. Mary Soames, who died at the age of 91, was his last surviving child.
A painting by Sir Winston Churchill has been sold for almost £1.8m at auction.