Having looked all but down just a few games ago, the Premier League's bottom club are now only three points from safety after following up Wednesday's 2-2 draw at Manchester City with a superb 2-1 comeback win on Saturday over another title challenger in Chelsea.
Sunderland still have a game in hand on the two teams above them and host fellow strugglers Cardiff City, West Brom and Swansea City before the end of the campaign - opening up a genuine possibility of survival after Poyet had stated that his men needed a "miracle" on the back of a 5-1 reverse at Tottenham earlier this month.
But after Fabio Borini's late penalty gave them a shock win at Stamford Bridge, Poyet believes that the momentum has swung.
"If we don't stay up now it will be a shame really," said the Sunderland boss. "On Wednesday at five o'clock (ahead of the Manchester City game) we were dead.
"Then we got a point at Manchester City, it should have been three. We got three today here to give us a great lift and a great opportunity to stay up."
Chelsea took a 12th-minute lead through Samuel Eto'o but Poyet's men produced a stirring fightback, with Connor Wickham adding to his midweek brace with a close-range equaliser before Borini converted from the penalty spot eight minutes from time after Cesar Azpilicueta had brought down substitute Jozy Altidore.
"We needed to fight, we needed to defend, we needed to stay in the game, we needed to make it difficult for them and we needed to take a chance or two," added former Chelsea midfielder Poyet.
"Overall, a team at the bottom of the Barclays Premier League coming here and playing the way we play I think is quite decent.
"But that's us, that's our season. Every time we play against the top teams we perform somehow in a better way than when we play the teams at the bottom.
"So I'm not surprised - It's nice. It was special for me as well, but we needed the points so it's very welcome."
The victory ended Jose Mourinho's 77-game unbeaten run in home Premier League fixtures as Chelsea boss and Poyet's counterpart gave witheringly sarcastic praise to referee Mike Dean and officials chief Mike Riley during a tempestuous and brief post-match press conference
Dean's decision to penalise Azpilicueta's challenge on Altidore to hand Sunderland their late spot-kick was the incident most likely to have riled Mourinho.
Poyet sympathised with the Portuguese's reaction, stating that he would have been in a similar frenzy had such a call gone against his team.
"My view from where I was, I couldn't comment, but I saw on the computer and I understand why we can be talking if it was or not," he added. "It's very difficult.
"I never, ever would expect to get a penalty here.
"It (Mourinho's reaction) is not too bad. If it was against me I would be the same.
"Emotions at key moments in the season - it could be the key moment for the title that one.
"Maybe I understand why he (the referee) gave it, especially the linesman - I'm not talking about the referee now.
"The angle where the linesman is, and when you see Azpilicueta going down trying to put the foot across, maybe he thought it was a different kind of contact to the one the television is showing.
"But I understand. If it was against me and I lose the game with a penalty I would be fuming, for sure."
