Lionel Messi missed a penalty but bagged a goal and two assists as the Catalans made it four wins from four in La Liga this season
Despite wasting numerous chances throughout the match, Barca eased to a fourth straight league victory that moved them back to the top of the table and continued their run of not conceding in the Spanish top-flight this term.
Neymar saw three opportunities go begging in the first half, before eventually finding the breakthrough in the 34th minute.
Loukas Vyntra then picked up a red card for tripping Lionel Messi in the area just before half-time and, although the Argentine sliced his spot-kick wide for the first time in his La Liga career, Ivan Rakitic stylishly ensured Barca held a 2-0 lead at the break.
Neymar was withdrawn just after half-time following a knock on his ankle, but his replacement Sandro Ramirez swiftly opened up a three-goal advantage.
Pedro and Messi then added gloss to the scoreline in the final 30 minutes, extending a run without a league defeat at Levante that began during the 1964-65 season and leaving their winless hosts rock bottom of the Primera Division.
Luis Enrique acknowledged in the build-up that Levante would likely set up with 10 men behind the ball and the hosts unsurprisingly invited pressure onto themselves from the start.
In fact, Levante should have been behind after just three minutes as Jeremy Mathieu received Dani Alves' cut-back but could only produce a tame first-time effort that deflected wide.
Barca continued to slice through Levante at will and Neymar went close twice before the half-hour mark.
The Brazilian first fired at Jesus Fernandez from 10 yards, before nodding just wide after beating the goalkeeper to Javier Mascherano's lofted pass.
Neymar finally got his name on the scoresheet in the 34th minute, however, as he latched onto Messi's fine throughball, skipped past Jesus, and coolly slotted into an empty net.
Barca's advantage grew seven minutes later as Vyntra received a straight red card for tripping Messi inside the area, but the No.10 skewed his spot-kick wide.
Nevertheless, the visitors did go into half-time with a two-goal cushion as Rakitic blasted a ferocious half-volley in from 25 yards, opening his Barca account in emphatic fashion.
The visitors endured a disappointing start to the second half as an ankle knock forced Neymar off after an innocuous clash with Pedro Lopez.
The 22-year-old appeared unhappy to be taken off but his replacement Sandro made an instant impact, stroking home from the edge of the area after being teed up by Messi in the 57th minute.
Barca's fourth arrived seven minutes later as Jordi Alba burst into the Levante area before handing Pedro a tap-in after drawing the goalkeeper out.
And there still time for Messi to make amends for his surprise penalty miss earlier in the game.
The forward capitalised on a dreadful attempted clearance 13 minutes from time to delicately chip in from the edge of the area, capping off a resounding victory for Luis Enrique's men.