Victory would have sent the Amsterdam heavyweights level on points with the Catalans in the battle for top-two spots in Group F.
However, despite taking the game to their illustrious opponents, especially during the first half, Ajax were undone by a Lionel Messi brace and consequently eliminated.
"I saw Ajax play like I wanted to see, similar to last season, pressing the opposition and playing good football," De Boer told a news conference. "It's a pity we could not reward ourselves.
"Our decision-making with the ball was good. We pressed Barcelona so well that they made many mistakes.
"I feel we deserved better. Not many teams can say that after playing Barcelona.
"In the future we must remember that matches are not won in one half but over an entire game."
