The Old Lady remain winless, with only one point from three games after their latest defeat.
Alvaro Morata swooped on a defensive error to open the scoring for Juve in the first half but Napoli found a way back.
Wojciech Szczesny couldnt hold Lorenzo Insigne’s curling 57th minute effort and Matteo Politano fired home the rebound.
Juve were clumsy in defence from an 85th minute as Szczesny didnt come, with substitute Moise Kean deflecting towards goal, before Kalidou Koulibaly bundled over the winner.
“Defensively we played very well today, unlike against Udinese and Empoli,” Allegri said.
“I have no complaints about my players. The game was decided by individual mistakes and moments.
“We obviously need to do better technically because we were a bit sloppy and should have scored another goal in the first half but we played with real character. Napoli deserve credit for the win.”
The pressure is mounting in Turin already, with Juventus finishing fourth last season, ending their decade-long dominance in Serie A.
Expectations are for immediate results with Allegri but he insisted the performance against Napoli was a “step forward”.
“At the moment we’re paying the price for every mistake but we’ll keep working away,” he said.
“We played well tonight and it was a step forward, but there’s not much you can say when you lose. It’s a long season so we just have to start winning, then stay focused so we can keep it up.”