The 16-year-old cancelled out Roberto Torres' opener just six minutes after coming off the bench, before fellow substitute Arthur buried a well-taken second goal for the defending league champions.
Torres sealed a point from the penalty spot after a Gerard Pique handball, though, with the result likely to add further pressure to Barcelona head coach Ernesto Valverde.
Valverde was again unable to call upon the injured Lionel Messi, Luis Suarez or Ousmane Dembele and left Ivan Rakitic on the bench.
The match was preceded by a minute's silence for Luis Enrique's daughter Xana, who died this week after a five-month battle with bone cancer.
Torres put Osasuna ahead in the seventh minute with the first shot of the match, thumping a volley past Marc-Andre ter Stegen at the left-hand post after being left completely unmarked.
It proved to be the only effort on target of the first half, as Barca produced next to nothing of note with their dominance of possession, although Jordi Alba felt he should have won a penalty.
Fati came on for Nelson Semedo at the break and Valverde's reshuffle paid dividends, the 16-year-old converting Carles Perez's cross with a fine header.
Barcelona were suddenly galvanised and it was 2-1 13 minutes later, Arthur cutting back onto his right foot and drilling past goalkeeper Ruben Martinez from just inside the area.
Brandon wasted a glorious chance to level when he made a mess of a free header, but Torres made no such mistake from 12 yards after Pique was penalised for blocking a cross with a raised arm.
What does it mean? Valverde pressure mounting despite faith in youngsters
Valverde has been criticised in the past - not always fairly - for a perceived reluctance to trust La Masia prospects within his first team.
With Messi, Suarez and Dembele injured and Antoine Griezmann not in the game, it fell to Perez and Fati to combine for the equaliser here.
The fact it was two youngsters that made a difference might just ward off some of the coach's critics, but results such as this will not keep the wolves from the door for long.
Perez a persistent menace
Having been the brightest player in an insipid first-half Barca display, it was fitting that Perez should set up Fati for his fine equaliser.
The 21-year-old is fast becoming an important player for the Catalans and should continue to be so if, as expected, Neymar does not return before the transfer window closes.
Griezmann goes missing
The service to Griezmann was not exactly of consistent quality, but this was a worryingly anonymous performance from the €120m forward.
Barca host Valencia on September 14, when they hope to have Messi available again. Osasuna travel to Real Valladolid a day later.