The result left the champions with only four points after three games.
Roberto Torres’ thumping volley gave Osasuna the lead in the seventh minute to send the boisterous El Sadar crowd into a frenzy.
Barca struggled to cope with the home side in a first half in which they failed to have a single shot at goal.
Coach Ernesto Valverde turned to Fati at the start of the second period and the teenager needed only six minutes to get off the mark, rising to head home a cross from fellow academy graduate Carles Perez.
Barca took the lead in the 64th minute through a superb strike into the bottom corner from Arthur Melo.
However, Osasuna were given a way back when Gerard Pique handled in the area and Torres made no mistake from the spot to equalise in the 81st minute.