The Argentine admitted that his side were distinctly second-best in an opening 45 minutes which saw the hosts dominate and race to a 2-0 lead with goals from Thulani Serero and Danny Hoessen.
"We didn't show our identity and we lost the match in the first half," he told Uefa.com.
"On an individual level we were not as good as Ajax. We lacked precision and mobility."
Despite being handed a lifeline with the early second-half dismissal of Joel Veltman - Xavi halved the deficit from the resulting spot kick - Barca failed to conjure an equaliser against 10 men.
It was Martino's first defeat in 21 games since taking charge in the summer.
Opposing coach Frank De Boer lauded his "lionhearts" as their dogged determination allowed them to cling on for a win, reviving their hopes of progress to the next round.
They are currently third in Group C on seven points, just one behind AC Milan.
Pointing to a first-half display brimming with attacking intent, the Dutchman said: "I think we put so much pressure on them they could not play their usual way. They are used to playing their style, but they lost a lot of the ball in midfield and had to depend on counterattacks."
"We want to go to the next round, but we need to win in the San Siro (against Milan) to do that.
"We will have to play another match like this one here."
Barcelona, who have already qualified for the knock-out stages, need at least a point in their final match against Celtic to guarantee top spot.