After goals deep in extra-time from Phil Bardsley and then Javier Hernandez had made it 2-1 on the night to United and 3-3 on aggregate in the semi-final second leg, the Black Cats triumphed 2-1 in the resulting penalty shootout to progress to a Wembley showdown with Manchester City.
Sunderland, currently in the Barclays Premier League relegation zone, have not been in a major final since 1992 and were last in the League Cup final in 1985.
Poyet said afterwards: "I would have been proud even with the 1-0.
"If the game finished 1-0 and we were out after extra-time, I was feeling it was my team playing all night, going on believing, taking risks, passing the ball around, trying to put Man United on the back foot.
"We kept going and going and going, so credit to the players - how they put an extra bit in to go to another level.
"After the (Bardsley) goal, I was not expecting to concede and it was difficult to pick them up for the penalties, but they looked very confident.
"I think it is a great night for a group of players and a group of fans that have been having a very difficult season."
The Uruguayan manager reserved special praise for Sunderland goalkeeper Vito Mannone, who saved two spot-kicks.
Asked about his team's achievement, Poyet said: "It is very difficult to explain.
"I don't remember another game like this one, being involved as a player or anything.
"Why? Because normally, when you score in the last minute or so you don't expect to then concede within one or two minutes, and then if you miss the first two penalties, you don't expect to win the game either.
"So how we did it? I think we need to ask Vito Mannone - I think he can personally answer that because he has been getting better and better and today he was the difference between the two teams."
He added: "When Vito got into the dressing room, the whole squad and staff started singing his name and there is no better feeling for a football player.
"He was half happy, half embarrassed. It was a special night for him."
Poyet revealed that his players had been so keen to step up for penalty duties that there had even been a heated dispute between two of them over who would take one of the spot-kicks - Bardsley and Ki Sung-yueng, who ended up scoring it.
Poyet said: "I couldn't believe it. The players were asking for the penalties. You don't get that too many times.
"We even had a little bit of a fight between Phil Bardsley and Ki for the fourth! We are going to have to practice more for the final, just in case."
In City, Sunderland - who fell behind in the first half of normal time through Jonny Evans' goal - have opponents in the final who are in formidable form.
The Black Cats will start the Wembley contest as heavy underdogs, but they did beat the Blues in the league this season and Poyet believes they are capable of defeating them again.
He said: "What a task it is going to be there at Wembley. We will try (to win).
"We will need to play well, have a good day, be brave, be fit and they will have to not have a good day, because they have very good players.
"But if you want to win something you need to be the best - you are not expecting to get to the final and play a bad game. So we will see.
"City are probably the most in-form team in the league, but it is just a game.
"You need to play well, play your best - to be spot on, practically perfect to win it.
"But it can happen.
"I just hope we don't have the other situation - being terrible and conceding four or five and suffering embarrassment."