The 17-year-old raced clear for the first goal, then fired home from an angle inside the penalty area.
Levante rallied in the second half but their pressure did not pay off until stoppage time.
Ruben Rochina’s half-volley from 25 yards took a deflection to wrong-foot Barcelona keeper Marc-Andre ter Stegen.
If Ter Stegen perhaps could have done better, the same can be said of opposite number Aitor Fernandez, who was twice beaten through his legs by Fati.
For both goals, Fati was fed by Lionel Messi, who first picked out a superb cross-field ball, then jinked across the face of the 18-yard box.
Levante beat Barca 3-1 defeat at home in November but have never taken a point from the Nou Camp.