The winning bid came from an anonymous buyer participating by telephone in Friday's auction at Christie's, the New York Post reported.
Christie's started the bidding at $500,000. Before the auction, there was speculation the price could rise as high as $2 million, the Post said.
Washington wrote the letter to Gen. John Armstrong April 25, 1788. Armstrong, who fought with Washington in the American Revolution, was a Pennsylvania delegate to the Continental Congress.
Chris Coover, the auction house's senior specialist on rare books and manuscripts, told the Post Washington was "heavily invested in the success of the new American Republic." Armstrong was also a supporter of the Constitution.
"It is one of the finest Washington letters we've ever handled," Coover said.