Two goals down following strikes from Steven Gerrard and Daniel Sturridge either side of the break, Poyet's men - who are three points adrift of safety third-from-bottom of the Premier League – looked down and out at Anfield.
But impressive substitute Ki Sung-Yueng headed home with 14 minutes remaining to set up a nervy finale.
Liverpool ultimately held on, but Poyet hopes his side can use their performance as a catalyst for victory against West Ham next time out.
"We did something nobody expected," he said. "The game was set up for a big party with four or five goals after half an hour.
"The idea was always to stay in the game. But we put Liverpool under pressure and it was tense."
He added: "Now it's West Ham (next Monday) and we need to somehow win the game. I am pretty sure (this will give us confidence).
"Not every team we will play has (Luis) Suarez, (Daniel) Sturridge and Steven Gerrard. We should be doing it (playing in that way) every week; now, you know we don't for some reason.
"Maybe we need to put another name to West Ham and say it's Real Madrid or Barcelona coming to the Stadium of Light. Then maybe we have the mentality that we can do something extra."
