The 45-year-old has agreed a two-year contract and will take charge with immediate effect.
Poyet, who left Brighton in the summer, will be joined at the Stadium of Light by his backroom staff Mauricio Taricco and Charlie Oatway.
"It's a huge challenge but one I am excited about and I am grateful for the opportunity to manage in the Premier League," Poyet said, via Sunderland's official Twitter feed.
"I look forward to proving my ability to the fans. I want them to believe because we need to stick together."
Sunderland chairman Ellis Short told the club website: "We analysed a wide range of candidates and believe that Gus's track record, experience, commitment and passion make him the right man to take us forward.
"We welcome him to Sunderland."
The Uruguayan began his coaching career in England in 2006 at Swindon Town, before joining Leeds United as assistant manager.
After a year at Elland Road, Poyet returned to his former club Tottenham Hotspur as first team coach, before becoming assistant manager to Juande Ramos.
In 2009 he was appointed as manager at League One Brighton and led the club to the Championship play-offs last season where they lost out to Crystal Palace.
Short also paid tribute to Kevin Ball, who took the role of interim head coach following the departure of Paolo Di Canio.
"I would like to thank Kevin, along with Robbie Stockdale, Mark Prudhoe and Scott Pearce, who stepped in at a moment's notice in what was a very difficult situation and gave 100% to the cause.
"As usual, Kevin gave his all for the benefit of our club and we are very grateful to him for his ongoing commitment."
Sunderland owner Short insisted at the weekend that staying in the Premier League was vital and great care would be taken to ensure the right man was brought in.
"Here at the club, like every supporter, we would like to see long-term success on the pitch and stability at the club," Short said in his matchday programme notes ahead of the clash with Manchester United.
"Unfortunately sometimes the quest for stability can be interrupted by the absolute necessity of staying in the league.
"This is because the long-term aim becomes irrelevant if we aren't at the top level.
"We now have a very important decision to make. Our only consideration in making that decision is what is best for the club. In doing so, we feel the weight of the entire city and beyond, along with the history of this massive football club.
"The appointment of a replacement head coach is our priority and I urge you to ignore the speculation in the media about the selection process, because most of it is completely wrong."
Ball admitted after the loss to United that he had 'no idea' when an appointment would be made, but did say that he felt the team had made progress since he took charge.
He said: "There are standards I have spoken to them about in training. They have worked hard, but there are still a lot of areas to improve on."