A 2-1 loss to West Ham on Monday left Poyet's side four points adrift of safety but they hold a game in hand over 17th-placed West Bromwich Albion.
The Uruguayan was appointed at the Stadium of Light in October last year, signing a two-year contract.
Poyet, 46, said he would offer his resignation if he saw his side giving up late in the campaign as he urged them to fight for their place in the top flight.
"If my team give up, I'll leave. Nobody needs to sack me, I'll leave on my own, OK?" he told a news conference after Monday's loss.
"If you see this team giving up before the end of the season, you won't see me here."
Goals from Andy Carroll and Mohamed Diame set up West Ham's victory, and they held on despite substitute Adam Johnson pulling one back for the hosts on 65 minutes.
Poyet said he was hurt and sad about Sunderland's position but believes repeating their recent performances could help them survive.
"If we play like we play against Liverpool and today in the next eight games, we're going to have a great chance (of surviving)," he said.
"The crystal ball, I can tell you, is not in my house. I'm not going to try and read the future because I'm not very good at that.
"What I believe is that I need to convince the players to keep doing it and defend this club to the end, to the last minute – to try their best.
"If everybody does the same, who goes out? The three worst teams in the league."
The loss extended Sunderland's miserable runs in Premier League games on Mondays.
The club are winless in 19 league matches on Mondays, dating back to 2002, and have a trip to Tottenham next – again on Monday.
"Thank you to the TV for putting another game on Monday next week. Very nice," Poyet said with a laugh.
"Why they don't change the last seven and put it on Mondays as well?"
