With only two games left to play ahead of their clash with Levante, Barca looked on course to register an "Invincible" season of their own.
Barcelona sewed up the La Liga title long ago, but the Catalan giants were chasing an invincible league campaign heading into the penultimate weekend of the season.
Ernesto Valverde, beaten in La Liga for the first time as Barcelona boss, may regret his decision to rest top scorer Lionel Messi for the trip to 15th-placed Levante.
It was 1932 when the last side, Real Madrid, went unbeaten in the top-flight but even that was across only 18 matches, in a 10-team division.
Barcelona rallied from 5-1 down on 56 minutes but failed to force the all-important equaliser at the Ciutat de Valencia. The unbeaten record was supposed to help atone for the European failure.
Boateng fired hosts Levante into the lead in the ninth minute with a simple tap-in.More news: Assam Govt Pacifies Protesters on Citizenship Bill
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And their position was improved further still less than five minutes later as Boateng became the first player this season to score a hat-trick against Barca after being played in behind the champions' defence and coolly slotting past the German goalkeeper.
Levante restored their two-goal lead barely 40 seconds into the second half in spectacular fashion.
Another counter just before the hour ended with another crisp Bardhi finish, as Barca were now staring at a four-goal deficit.
1 - Coutinho is the first player to score a hat-trick for Barcelona and finish on the losing side in the 21st century. The Brazilian grabbed his hat-trick, firing in a shot from outside the box to make it 5-3 before Suarez tucked away a penalty for Barca's fourth after Boateng was penalized for tripping Sergio Busquets in the box.
But without Messi even to bring on, Levante withstood the late onslaught and should even have scored a sixth only for Ruben Rochina to prode wide after a terrible backpass by Busquets.
After dribbling his way into the opposition's final third, the centre back handed the ball to Coutinho, whose right-footed strike from the edge of the area took a wicked deflection before nestling into the back of Oier's net.