Three league wins in a row has lifted Poyet's team off the foot of the table and two points above the drop zone, with a game in hand over 18th-placed Norwich and a seemingly insurmountable goal difference.
They are on the brink of a superb escape but, after witnessing Liverpool's sensational collapse at Crystal Palace on Monday, Poyet insists his team must ensure there are no costly final slip-ups.
"The idea is to be concentrated on us in the way we have been in the last few games, to not change anything," he said. "We want to win the game because I don't know how to play for a draw or to get a point.
"A point would be enough, not mathematically, but realistically because if we lose the last game 7-0 and Norwich beat Arsenal I would think something strange is going on in those two games.
"But we saw yesterday what can happen in 15 minutes to anyone, even the best teams in the league.
"Anything can happen on the day so we need to do the job, start aggressively, connect with the fans, put pressure on West Brom."
Sunderland dropped to the bottom in early April and were seven points adrift of Norwich before a draw at Manchester City and wins over Chelsea, Cardiff and Manchester United gave them 10 points from a possible 12.
"We've been in a bad period for a long, long time and the word then was confidence. We couldn't get a goal and couldn't get a break because of confidence," Poyet said.
"But now we're on the other side and it's exactly the same, we have big confidence but we need to play, defend well and put pressure on West Brom and then our confidence will help us win the game."