"I would like to thank (chairman) Ellis Short and the board for giving me the opportunity to manage Sunderland and the fans for always being so passionately supportive of their club," Moyes, 54, said in a statement.
“I wish the players and my successor well in their efforts towards promotion back to the Premier League”.
Short added: “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.
“Having worked tirelessly throughout the campaign to avoid relegation from the Premier League, David has chosen to leave the club without compensation, which is testament to his character."
Sunderland appointed Moyes on a four-year contract in July to replace Sam Allardyce who left to become England manager.
However, the former Everton and Manchester United boss, who had been released by Real Sociedad, struggled at the Stadium of Light. Sunderland finished bottom of the table with 24 points after only six victories in 38 games, scoring 29 goals.
The Scot, who was sacked by Real Sociedad after just a year in charge, had last managed in England during an ill-fated spell at United in 2013-14, following Alex Ferguson's retirement.
Moyes had started his managerial career at Preston North End, winning the third-tier crown, before leaving to take over at Everton where he spent more than 11 years.
He was Sunderland's fourth full-time managerial appointment since Italian Paolo Di Canio was sacked in September 2013.
The club had flirted with relegation in each of the previous four seasons before finally going down this term.
Former Sunderland striker Kevin Phillips is the bookmakers' favourite to take over with former Middlesbrough manager Aitor Karanka and Leeds United boss Garry Monk also backed.
Chairman Short added: “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.”
(Reporting by Hardik Vyas in Bengaluru and Ken Ferris in London; Editing by Alison Williams)
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