Having made a spectactular save from a Marek Hamsik header in the first half, the Chilean, who signed from Real Sociedad in June, flapped at a Blerim Dzemaili shot from distance late on to concede the only goal of the game
The strike came against the run of play after Barcelona had dominated possession for much of the encounter in Switzerland, and could harm Bravo's chances of challenging fellow new boy Marc-Andre ter Stegen to be Luis Enrique's first-choice custodian.
Barcelona - without the likes of Lionel Messi and Neymar, who underwent medicals in Spain on Wednesday following their prolonged World Cup campaigns - handed midfielder Ivan Rakitic his first start for the club.
Napoli striker Lorenzo Insigne had a glorious chance to take an early lead against the Catalan giants.
Hamsik found the Italy international with a superb ball from the right, but Insigne appeared to be caught in two minds as the ball bounced harmlessly beyond him in the penalty area.
New recruit Bravo was soon called into action as he leapt superbly to his left to tip a looping Hamsik header wide of the post.
Despite all their possession, Barcelona's attacking threat was limited throughout the first half against a solid Napoli defensive unit, but Rafael Cabral got down well to deny Munir El Haddadi seven minutes before the break.
The Catalan giants began to turn the screw early in the second period and Pedro came close to opening the scoring on two occasions - first seeing a low drive touched wide by Rafael, and then sending a left-footed effort narrowly past the other post when he should have done better.
Both coaches made full use of their benches and by the 75th minute only three players who had started the match remained on the pitch.
One of those players was Bravo, but he was left wishing he had been withdrawn when he let Dzemaili's half-volley slip through his fingers 10 minutes from time.
Rafael got down well moments later to deny Gerard Deulofeu an equaliser from a curling effort, as Barca were unable to come up with an equaliser.
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