Head coach Poyet stated after last month's 5-1 hammering at Tottenham that it would take a "miracle" for his side to avoid the drop, but they now have control of their own destiny.
A Sunderland victory at Manchester United on Saturday, combined with a Norwich City defeat at Chelsea the following day and both Cardiff City and Fulham failing to win, would all-but ensure the Wearside club remain in the Premier League.
And Poyet, whose side have a significantly better goal difference to the three sides below them and a game in hand, believes it would afford him a first opportunity to relax since taking the reins at the Stadium of Light in October.
"That would be the first week during my time at Sunderland that I could relax a little bit, maybe play golf," he said.
The Uruguayan is reluctant to state how many points he reckons would guarantee his side's survival, but is eager to avoid a tense final day.
"It's very difficult to say," he continued. "Our idea a long time ago was four wins, so we are still two wins away.
"We need to make the same points as Norwich and hope that the other two teams do worse than us.
"The closer you get, it's not about the points, it's about having three teams behind you somehow.
"The best way is winning a couple of games.
"It could go to the wire, there is still a possibility, so we need to make sure we take our opportunity this weekend."
Connor Wickham was handed the Premier League player of the month award for April and Poyet expressed his delight for the striker, who has scored five goals in his last three appearances after spending loan spells at Sheffield Wednesday and Leeds United this season.
"He totally deserved it because, with respect to everybody, there wasn't a player in that kind of form in the last few weeks," Poyet said. "There were teams that did very well - Palace won four or five games in a row, but there was not one player standing up and being the difference.
"But for us, it was Connor with no doubt."
However, Poyet stopped short of tipping Wickham for a place in England's FIFA World Cup squad, adding: "Every single English player needs to be ready. I know a few have more of a chance than others."
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