Thulani Serero and Danny Hoesen fired Frank de Boer's men into a 2-0 half-time lead, before Joel Vetlman was issued a red card which led to the Spain international reducing the deficit from the penalty spot.
Despite the numerical advantage, the Catalans failed in their bid to find the equaliser and, speaking to reporters at full time, Xavi congratulated the Eredivise champions on their Barca-type performance.
"In the second half, we were better. Against 10 men we dominated and we had some chances to equalise," he told reporters. "Sometimes, in two or three moves, the opposition can open you up at the back and afterwards you lose fluidity."
"I'd like to congratulate Ajax because, above all in the first half, they were more intense than we were. After the break, you saw a different Barca.
"They played in a way in which we normally do and it was difficult to get the ball off them."
Despite the defeat the Blaugrana remain top of Group H in their Champions League group, two points ahead of AC Milan in second.
However, victory over Milan in the sixth and final gameweek would see Ajax trump the Italians to a spot in the latter stages of Europe's elite competition.