Martino's first Champions League match in charge ended in a resounding success at the Nou Camp, with Lionel Messi netting a hat-trick.
Martino said on the club's website: "FC Barcelona recently reached footballing excellence and therefore the fans are more demanding.
"I understand that they didn't like the first half, that they weren't satisfied. But they should be fairly satisfied with the second half."
The Catalans won the Champions League twice in three seasons under Pep Guardiola, most recently in 2011.
Martino said his objective was to "reach the same place Barca always reach" but he will not stick exclusively to a tiki-taka style.
He added: "We don't want to change the team's style of play, but I saw (former Barca defender Rafael) Marquez make long passes too five years ago.
"It doesn't mean we're changing the style if we add alternatives to our play."
The Argentinian accepted his side's first-half performance against the Amsterdammers was not up to scratch, but was happier with the way they played after the break.
"It wasn't a brilliant match, but we put in a strong performance," he said.
"Our second half was better than our first. We controlled the game better, we pressured and we defended well.
"If we had played the second half like we did the first we would have taken a step back, but we corrected our errors and I believe that we took a step forward.
"We had a noticeable change. I think our victory is just, especially because of what we did in the second half."
Ajax boss Frank de Boer saw positives from his side's start to the game - they trailed only 1-0 at the break - but said they were punished in the second half.
"I'm very pleased with the first half," he said on the Ajax website. "We had just as many opportunities as Barcelona, and continued to play good football.
"I thought that we gave away too much space in the second half. In the end, it cost us four goals, and that's painful. We need to stay positive and look ahead to the next game."