The wartime prime minister abandoned the stub to attend a Cabinet meeting - and it was picked up by a Downing Street worker.
The valet sent the four-inch stub to a friend with a note jotted on Number 10 writing paper.
The friend kept the souvenir until his death, at which point the souvenir stogie passed to a relative.
It was bought by a private collector at a sale in Norfolk.
Auctioneers Keys, of Aylsham, said the cigar had been expected to fetch around