Fabio Borini's opener ensured that Manuel Pellegrini's side trailed at half-time, but an inspired display after the break saw them turn things around to run out 3-1 winners thanks to fine strikes from Yaya Toure, Samir Nasri and then substitute Jesus Navas.
Kompany captained City to their first silverware of the season and he was delighted with how they bounced back following a frustrating first half.
"I thought it was a good performance in the second half, we were the better team in the second half," he told Sky Sports.
"We came out and showed some really good stuff and the goals we scored were unbelievable. We showed quality but obviously not in the first half."
The win helped City to erase memories of last season's FA Cup final defeat to Wigan and Kompany spoke of how special it was to lift a trophy at Wembley.
"It's the best stadium for me," he said. "It's the best stadium in Europe, even in the world maybe when it comes to football venues and just lifting the trophy here is something special."
City are still chasing silverware on three other fronts this season but Kompany wants them to take things one step at a time.
He said: "We never take anything for granted and we will enjoy every moment.
"It's a first (trophy). I think we should be hungry for more but it's nothing more than that. We've achieved something great for the club and we need to carry on."
Speaking of Manuel Pellegrini's influence, Kompany added: "He's obviously brought us a certain style of playing the game and mostly I think our fans enjoy it. If we win trophies like this, we will enjoy it."