Burnley secured automatic promotion to the Premier League with seven games to spare after Friday's 2-1 victory at Middlesbrough ensured Kompany's side will finish in the second-tier Championship's top two.
They make their return to the top flight after one season, having been relegated on the final day of the 2021-22 campaign, in which they finished third-bottom.
Kompany, appointed in June 2022, said he had not dreamt of this outcome.
"It's Easter, there are seven games to go and we are already celebrating," Kompany told Sky Sports. "We didn't expect this. We wanted to experience it one day, but we had a different timing on it. Quicker is better sometimes.
"Days like today happen more than you think, we found a way it was not easy at all. But somehow this season we also ended up on the good end of the game.
"There is a belief in the team, and it is still a team that can improve and that is the exciting part. They're like kids, they're celebrating like kids and that's fun to see."
Club chairman Alan Pace said the promotion came as a "complete surprise".
"Vincent and I spoke during the summer and talked about it in great length. We gave ourselves two to three years ... What you're seeing is a lot of magic coming together," Pace added.
Burnley need 11 points from their last seven games to win the Championship title, while 13 points will see them become the first Championship team to rack up more than 100 points since Leicester City in 2013-14.
