Lionel Messi, who returned to the Argentina squad this week, made it six goals in his past five games with a first-half penalty that appeared to have leader Barca on course to extend a seven-match winning streak in all competitions.
But after Luis Suarez went off injured, an astonishing spell saw Jose Campana, Borja Mayoral and Nemanja Radoja all score as the match was turned on its head.
After a quiet start to the contest, Antoine Griezmann saw a right-footed effort saved by the outstretched leg of Levante keeper Aitor Fernandez, Messi's throughball having created the opportunity.
Barca's breakthrough arrived seven minutes before the break, a slick passing move ending when Arthur found Nelson Semedo, who was hacked down by Jorge Miramon in the box.
Messi stepped up and comfortably sent Fernandez the wrong way from 12 yards, finding the top-right corner.
He almost had a second with half-time approaching, only for a combination of Fernandez and Miramon to deny him from close range, while Barca was forced to replace an injured Suarez after the striker could not run off an earlier knock.
Levante did not produce a shot on target in the first hour, but levelled the match when a poor Gerard Pique clearance led to Mayoral and Jose Luis Morales combining to set up Campana, who confidently beat Marc-Andre ter Stegen from just inside the area.
Just two minutes later, Mayoral collected a pass from Campana and curled an excellent 20-yard strike beyond Ter Stegen, who got a hand on it.
An astonishing sequence concluded with Levante's third after 68 minutes, with Clement Lenglet's headed clearance volleyed in by Radoja from 18 yards, courtesy of a big deflection off substitute Sergio Busquets.
VAR (video assistant referee) ruled out a second goal for Messi because of an offside decision against Griezmann in the build-up, and that was as close as Barca came to producing a fightback, despite the arrival of Ansu Fati from the bench.
The injury to Suarez afforded Griezmann the chance to play more centrally, but he struggled to make an impact as his mixed start to life at Barca continued.
He had a first-half chance saved, found the side netting before being spared by the offside flag, and did not create a single chance for his team-mates.