A Jonny Evans goal in the 37th minute looked to be enough to send Manchester United through on away goals at Old Trafford, but Phil Bardsley netted a late leveler in extra time to put Sunderland 3-2 ahead on aggregate.
Javier Hernandez capped a frantic last two minutes by netting another for United to send the match to penalties, which saw only three players find the net before Vito Mannone made a crucial save from Rafael to seal a 2-1 spot-kick victory, booking a March meeting with Manchester City at Wembley.
"You know how difficult it's been this season for us," he told reporters. "It's been really tough for the players, the fans, the club, the chairman. Every single one.
"To have a day of football like today is incredible.
"It's great for everybody, great for football. I know for Man United it's not that good. I thought it was a great story for football - let's see if we can go one more.
"The things that happened today are incredible, but the players have been outstanding. Credit to them."
Asked whether he thought Sunderland could defeat Manchester City in the final, Poyet looked ahead to what he knew would be a tough challenge.
"Who knows?" he added. "It's going to be an incredible task. We need to be at our best, hope Manchester City have a bad day, be very organized and take our chances.
"The idea is to go and make the fans proud as well, so if we play like we did today, maybe we get that chance."