The Stadium of Light outfit stole ahead at Wembley through Fabio Borini's superb strike with the outside of his boot, to evoke memories of City's disappointing 1-0 defeat to Wigan Athletic in May's FA Cup final.
City stepped up a gear in the second half, however, and were ahead by the 57th minute courtesy of unstoppable strikes from Yaya Toure and Samir Nasri.
Jesus Navas rounded off the win in the closing stages, meaning Sunderland's wait for a first major trophy since they collected the FA Cup in 1973 continues.
Poyet was naturally disappointed, but feels his side can take confidence from their performance as they switch focus to an ongoing battle against relegation from the Premier League and an FA Cup quarter-final against Hull City.
"We gave it a go, but the chance to score a second goal, maybe that was the difference," he told Sky Sports.
"We knew everything had to go for us, we knew it. Yaya Toure scored a wonder goal, Samir Nasri scored an incredible goal too. We did everything we could.
"That's the best part of what we are all about. To beat us they had to score two wonder-goals. I hate losing but I'm proud of the players.
"If we do this every week I'm sure we will stay in the Premier League and will come back to Wembley (in the FA Cup).
"I've said from day one, it was nearly an impossible job to stay in the Premier League. We were at the bottom, too far from the rest but we have done some special things and I hope this gives us the experience to stay in the Premier League."
