Levante rallied in the second half but their pressure did not pay off until stoppage time.
The first half was one-way traffic as Messi poked an early effort wide before bringing a save from Fernandez Aitor after surging towards goal from the right-hand side.
Antoine Griezmann opened the scoring for Barcelona in the 4th minute. The teenager brushed off the challenge of Jorge Miramon before stroking the ball into the net.
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.
Tottenham complete the signing of PSV Eindhoven forward Steven Bergwijn
Since then he has made 149 appearances for the club, winning three Eredivisie titles, while contributing 31 goals and 41 assists. A PSV statement read: " Steven Bergwijn did not join his PSV team-mates for training on Tuesday morning".
Less than two minutes later, Fati was in again. Arthur scored in the 77th minute before Messi netted his second goal of the day in the final minutes of the second half.
Key performances from Nelson Semedo, Clement Lenglet, Arturo Vidal made the difference last night which in turn gave more chances to Messi and Griezmann to score against the opponents.
Fati nearly completed his hat-trick in spectacular fashion midway through the second half but his overhead kick from Frenkie De Jong's cross went narrowly wide.
According to Barcelona website, it took 710 games for Messi to reach that mark after his team bounced back from a 2-0 defeat to Valencia in La Liga over the weekend.
Instead, Setien has started Fati in all five of his games in charge so far and this was the first time his faith has really been rewarded.