Barça had won 16 and drawn the other four of their 20 competitive matches since Martino took the reins this summer, and they had the chance to equal a club record of 21 games unbeaten at the start of a season in Tuesday's match in Amsterdam.
That previous record was set by Pep Guardiola's side at the start of the 2011/12 campaign, but the current crop fell just short as they were beaten 2-1 by ten-man Ajax. Although the loss was not a costly one as Barcelona had already secured their place in the next stage of the Champions League, Puyol knows they need to improve.
He told the club's website: "You have to learn from every game and if we want to win things, we know we have to get better."
Xavi converted a penalty for the visitors to halve the deficit at the start of the second half but, even though Ajax were reduced to ten men in that incident with Joel Veltman being sent off, the Catalans could not find an equaliser.
Xavi said: "We hope it doesn't affect us too badly. It's the first defeat and let's hope it won't have too bad an effect."
Martino demands improvement
Barça will now turn their attentions back to La Liga, where they will aim to get back to winning ways and try to extend their three-point lead at the summit when they take on fifth-placed Atletico Bilbao on Sunday.
Martino said: "We all have to reflect on this and think about the difference in intensity we saw out there. We can't allow ourselves to play like we did in the first half."