Luuk de Jong doubled Barca’s lead scoring his first goal in his fourth appearance for the club after his Netherlands team-mate Memphis Depay opened the scoring from the penalty spot.
An inspired performance from Aitor Fernandez prevented Barca from significantly boosting their goal difference as they totally dominated the game, with under-pressure boss Ronald Koeman watching on from the stands as he serves a two-match ban.
Fati, on as a late substitute, put the icing on the cake in his comeback following 10 months out with a knee injury, finishing superbly in stoppage time as Barca ended a run of three games without victory.
Depay sent Fernandez the wrong way from the spot just six minutes in, having produced great trickery to beat two defenders before being upended by Nemanja Radoja.
De Jong opened his Barca account eight minutes later, finishing clinically with his right foot after Sergino Dest had slipped him in.
Gerard Pique somehow failed to add a third from point-blank range and Gavi was unable to lob an advancing Fernandez, who showed great reflexes to tip the lively Depay’s header over the crossbar.
Levante were unable to contain Depay and Fernandez diverted his right-footed drive around the post after some shambolic defending.
Teenager Nico Gonzalez, making his first Barca start, flashed a venomous long-range drive wide and Fernandez thwarted Depay yet again.
Sergio Postigo almost turned the impressive Gavi’s cross into his own net before Fati was given a great ovation when he came off the bench with 10 minutes to go.
The 18-year-old looked like he had never been away as he spun away from a defender then beat another before finding the back of the net with a right-footed shot from outside the area.