Sunderland appeared all but relegated after a nightmare run that saw them take just one point from a possible 27, but a 2-2 draw a Manchester City on April 16 kick-started a remarkable revival.
Poyet's men won at Chelsea three days after their draw at the Etihad Stadium, before then beating now-relegated Cardiff City 4-0 last time out and winning at Old Trafford on Saturday.
The Wearside club are now three points clear of third-bottom Norwich City - who visit Chelsea on Sunday - with two games remaining.
"I'm feeling great," Poyet told reporters. "Our idea was always to win four games - we've won three, we've drawn one. We're still one win away.
"Of course we can have a hand from Chelsea tomorrow. We'll be watching but we need to concentrate on what we do. We're in a position where have two home games - win one and we're over the line.
"But what we’ve done in the last four games is unique, special. Maybe you’d even call it a miracle. Miracle is a word I used a few weeks ago. Apparently every now and again, it happens.
"We were down, down, down, down. We needed something special. After we lost against Everton we needed to win four games. We had City, Chelsea and United away. Expecting to win the three at home, we still needed one special result.
"The determination of the players has been incredible, the willingness to do things right. I said a long time ago, 'We're going to fight, whoever we play'. Today we showed that again.
"We've been down [in the relegation zone] all season. I said a few days ago, 'Another win and we stay up'.
"With 20 minutes to go we saw [Robin] van Persie and [Danny] Welbeck coming on. We were tired. Somehow we're going to have to defend.
"And we did it impressively. They practically didn't have a chance after that. We were fighting for our lives."