Barca's 3-0 second-leg win sees them reach the final 3-2 on aggregate.
"We kept fighting through the exhaustion, it was incredible," Koeman said. "I am so satisfied with the team tonight. We deserved this win. I think we were better in both games.
"We pressed high. The team is strong physically and mentally. We always believed (in the comeback). We've never chucked in the towel on the Copa. It's a question of mentality. As the coach, I can't ask more from my team than what I've seen tonight."
Asked if it had been his happiest night as Barça coach, he added: "I think so -- and also for the players."
He continued: "The team's growing with the changes we've made, giving chances to really young players. We have improved during the season. The team's good with the ball, we know that, but above all we have improved without it. We're stronger, we run more and we press more. I think we have good things ahead of us."