The Dutch champions trailed 1-0 at half-time following Lionel Messi's thunderous free-kick in the Group H fixture, which marked the first competitive meeting between the two sides.
Messi added a second in the 55th minute and Gerard Pique made it three before the Argentina international completed his hat-trick to hand Gerardo Martino a comprehensive victory on his Champions League debut.
And De Boer, a former Barca player, was left to rue Ajax's inability to build on a solid first-half display at Camp Nou.
"I was satisfied with the first half," he said. "We almost created as many chances as they did.
"It is a shame that very quickly in the second half we conceded the second goal.
"We also gave away two goals at the end. That is what really annoys me."
