The 45-year-old Uruguayan, a former Chelsea and Tottenham midfielder, ended his first managerial job with Brighton in controversial circumstances back in May but is back with a two-year contract at the Stadium of Light.
He told a press conference of his time at Albion: "My aim was to try and prove that I'm good enough for the Premier League and so here I am. I've got my chance now. It's going to be a difficult challenge, but I'm ready for it.
"When you're at the bottom in the situation that we are, everyone will say it's going to be difficult. The idea here is to believe and to be convinced. From what I see, I do believe that we can get safe.
"I need to convince the players and everybody. I need to make sure that players, staff, fans and directors believe in this opportunity that starts today. I'm going to play my part. I need to make sure everybody commits to the cause.
"You can see that they want to show everything the first day. Everybody wanted to train - even the ones that I wasn't expecting to. I just asked them to pay attention. The next two or three weeks, there's going to be plenty of information both ways.
"I need them to be really aware and to try to understand, to ask questions and to be involved. That will make it quicker. The longer it takes, the more difficult it is going to be. Most of the work now will be mental. It's about getting the right information.
"I've had one job as a manager so far and I was in it for four years. Every time a new manager arrives, he wants to stay as long as possible and prove he was the right choice.
"I am a confident, positive person but it's too easy to talk in advance. I want to prove it. I want to show you. When we get to the end of the season, I hope we are going to be sitting here smiling and I hope I can show I can do it at this level."
With the Black Cats rooted to the foot of the table having taken just one point from their opening seven games, Poyet's first game in charge will be the trip to face Swansea on October 19.
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.
He will be joined at the Stadium of Light by his backroom staff Mauricio Taricco and Charlie Oatway.
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."
Short also paid tribute to Ball, who took the role of interim head coach following the departure of Paolo Di Canio.
He said: "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."