The end of the Black Cats’ 10-year stay in the top flight was confirmed when they lost to Bournemouth last month.
Moyes, 54, informed chairman Ellis Short of his decision to step down at a meeting in London on Monday.
“I wish the players and my successor well in their efforts towards promotion back to the Premier League,” said Moyes.
Former Everton and Manchester United boss Moyes took charge in July last year, after Sam Allardyce left to become England manager.
Sunderland finished bottom of the table this season with 24 points, having won only six games.
“I pursued the services of David Moyes for a considerable period prior to his appointment last summer, which makes the announcement of his departure difficult for everyone concerned,” said owner Short.
He said that Moyes was not taking compensation for his departure, calling it a “testament to his character”.
“In the days ahead we will take some time for reflection, and then focus on recruitment and pre-season as we prepare for our Championship campaign. We wish David well in the future,” added Short.