The new manager says he hopes "we will be sitting here smiling" at the end of the season and must convince his players they are good enough to beat the drop.
The Uruguayan was unveiled as Paolo Di Canio's successor on Tuesday after the Italian was sacked last month. Poyet takes over Sunderland with the club currently bottom of the table with just one point from seven games, but the new manager believes the club can avoid relegation.
"I hope at the end of the season we will be sitting here smiling,” Poyet told reporters. "When you are at the bottom, everyone will say it will be very difficult. The idea here is to believe, to be convinced.
"From what I see, I do believe we can get safe. Now, I must convince the players.
"We must all understand that we really have to believe in this opportunity we start today. I will ask for help, as you can imagine, but it is my job to get everyone to commit."
The 45-year-old Poyet is also confident he possesses enough experience to be a success in his new role and is hoping for a long stay at the Stadium of Light.
"I had four years in my one job in management so far at Brighton. That speaks for itself,” Poyet said. "Every manager wants to stay as long as possible. I am confident, I am a positive person."