The Dutch champions became the first team to beat their Spanish counterparts this season on Tuesday night, claiming a 2-1 victory at the Amsterdam Arena.
The result meant that the group remains alive until the last round of matches has been played, with Ajax and AC Milan still gunning for a ticket to the knockout phase with Barca.
Ajax also thwarted Barca boss ‘Tata’ Martino’s bid to equal and then surpass the record of undefetaed matches achieved by former manager Pep Guardiola.
Even though they are safely through to the next round, Pique is aware that if Barca play like they did against Ajax when they face better teams in the future, winning the Champions League may prove impossible again this season.
"We played a horrible first half," the defender admitted.
"There is no excuse, it was a disaster. What happened in the first half can't happen again.
“The attitude wasn't right, the whole team know that."
Skipper Carles Puyol echoed his defensive partner’s negativity, and conceded that Ajax were the better team on the night.
"Ajax were a lot more intense than us," he said.
"In the first half they were far better, and you can't give one half away in Europe."
Ajax meanwhile, celebrated the goals scored by South African Thulani Serero and Danny Hoesen long into the night.
And their manager, Frank De Boer, is now confident that they can make it into the last-16 by beating jittery AC Milan in the final round of group games.
"I don't think Barcelona expected this, we have achieved a victory thanks to our fighting spirit and our quality," he said.
"The first half was fantastic, after that it was unfortunate that we were down to 10 men.
"I had said that Ajax would be the first team to beat Barcelona with the football we always want to play, which is offensive football."