The 45-year-old Uruguayan has been handed the task of lifting the Black Cats from the foot of the Barclays Premier League table following his appointment as Paolo Di Canio's successor on Tuesday.
Having signed a two-year deal, Poyet is confident he can repair the damage wrought during the Italian's ill-fated reign and ease them out of the relegation zone.
Asked if he had a message for the club's supporters, he told SAFSee: "Trust me, believe, connect with the team. We are all together in this, we are all together, so it is important that we stick together.
"It's a big, big challenge, but I am really excited. I thought I would have an opportunity in the Premier League and now I have got it.
"I need to make sure that I prove they picked the right man to get us from the situation we are in, so I am absolutely delighted."
Poyet inherits a team which is yet to win a Premier League game this season and has managed to collect just a single point from seven attempts to date.
He will look to address that situation for the first time at Swansea next Saturday before heading into his opening match at the Stadium of Light, a derby clash with Newcastle which has now taken on added significance for a man who twice guided former club Brighton to FA Cup victory over the Magpies.
Asked what his priority would be, he said: "Well it is to address the problems quickly, make sure we train and we convince the players to get better, and slowly that will make us pick up points.
"The sooner we win, of course, the better, but the idea is to make sure that we understand the way that we would like them to play football.