"I am delighted to have joined Sunderland. I am relishing the challenge and excited by the opportunity," Moyes said in a statement on Saturday.
"I have taken over a big British club with great support and I'm looking forward to working in the Premier League again. I look forward to continuing the good work done by Sam."
The Scot's last managerial job in England was his ill-fated spell in charge of Manchester United in 2013-14, following Alex Ferguson's retirement.
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Moyes has also managed Preston North End, winning the third-tier crown with them, and Everton where he spent more than 11 years.
He is Sunderland's fourth full-time managerial appointment since Italian Paolo Di Canio was sacked in September 2013.
The club have flirted with relegation in each of the last four seasons.
Uruguayan Gus Poyet took over from caretaker manager Kevin Ball in October 2013 and guided the club to 14th place in the league.
Poyet was dismissed in March 2015 with Sunderland one point above the relegation zone. The club appointed Dutchman Dick Advocaat who guided them to the safety of a 16th-placed finish with one game remaining.
Advocaat resigned in October 2015 with Sunderland second from bottom. Allardyce was appointed the same month and performed wonders to avoid the drop by ending up fourth from bottom.
"We are absolutely delighted to welcome David Moyes who was by far and away our first choice," Sunderland chairman Ellis Short said. "He is a man whose football pedigree speaks for itself and is someone I have long admired.
"He was my number one managerial target for the last five appointments but his desire to honour existing contracts meant we were not able to bring him to Sunderland previously.
"To be able to finally welcome him as our manager is fantastic news for the club," added Short.
Sunderland launch their new league campaign at Manchester City on Aug. 13.
(Reporting by Simon Jennings in Bengaluru; Editing by Sudipto Ganguly and Tony Jimenez)
