Midfielders Frenkie de Jong and Riqui Puig struck the goals for Barca.
"I saw a Barça side switched on from the very first moment and with a lot of possession of the ball," the Dutch coach said. "The key to the game was the work we did without the ball."
Koeman appreciated the capacity of his team to win the ball back in midfield, although he lamented a failure to take advantage of that.
He appears to have settled on a midfield trio of Frenkie de Jong, Sergio Busquets and Pedri, which he says is helping the team be more consistent, although Busquets will miss the next game.
"It's true, we know he has five yellow cards now (and will be suspended against Athletic Club in the league) but first we have the Copa del Rey game against Rayo Vallecano."
However, despite being generally satisfied, Koeman said that "there were 10 or 15 minutes when we dropped the intensity and there was a big mistake in defence, at least we have a great goalkeeper," in reference to Ter Stegen. But he also praised the performances of the defenders.
After De Jong opened the scoring, Riqui Puig wrapped up the win in the last minute. Koeman said the young midfielder "came on and did well, getting forward. He's not very tall but he jumped really high to score the decisive goal."