Kompany had only made 12 appearances for City this season ahead of the match due to a number of injury issues, but the Belgium international was given the nod to start against the Gunners.
The centre-back has branded the victory "a massive result", and claimed that he was confident ahead of the match that he would score.
"It was a difficult game, but as much space there was for us and Arsenal, we did well and controlled the space and I am proud of the team. Winning 3-0 at Wembley is a massive result against any opponent," Kompany told Sky Sports News.
"Before the game, I thought I was going to score. I came close during the game so there was some luck on the day, but also self-belief. I have been in the game for 15 years and I have been training well and supporting the lads when I don't play. When I get the call I want to do the job."
City have now won the League Cup in three of the last five seasons.