It is now 28 games since Barcelona tasted defeat in any competition, while they have won their last five in the league to take control at the summit, still with a game in hand.
Levante, meanwhile, remain bottom of the league on 17 points, but head coach Rubi will have been greatly encouraged by his team's performance against the reigning champions.
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Barcelona could have been 1-0 ahead inside the first minute when Lionel Messi found Luis Suarez , but the striker's effort was stopped by Diego Marino. The ball dropped to Andres Iniesta, but his effort was over.
Less than 30 seconds later the visitors did have the ball in the back of the net through Messi, but the flag was raised, even though replays demonstrated that the Argentine was onside when the initial pass from Iniesta was played.
Levante hardly had a touch of the ball inside the opening 10 minutes as the visitors utterly dominated, but the hosts had luck on their side in front of a vocal home support.
Rubi's side had a chance of their own in the 12th minute when Tono set up Deyverson, but the Brazilian's effort was just wide of the near post, before Neymar tested Marino down the other end during what was an entertaining 20 minutes of football in Valencia.
It should have been 2-0 less than five minutes later when Neymar set up Suarez with a low cross from the left, but somehow the in-form striker missed the target from close range.
Levante had their moments, however, and came close to a leveller in the 30th minute when Verza searched for the top corner from a free kick, but his effort was just too high, before Jefferson Lerma missed the target after breaking through on Claudio Bravo's goal.
Levante then came even closer in the 38th minute when Jose Luis Morales hit the post after breaking through on goal. Barcelona might have led at the break, but the hosts were impressive for long spells of the first 45 minutes.
The first opportunity of the second period also fell to the home side, but Giuseppe Rossi, on his home debut, saw his strike beaten away by Bravo.
Messi again had the ball in the back of the net for Barcelona in the 54th minute, but this time the flag was correctly raised as it remained 1-0 to the visitors. Ivan Rakitic was next to come close with a header just before the hour, but Marino made a smart save.
Levante boss Rubi threw on Nabil Ghilas in the 69th minute in a bid to poach a decisive equaliser and it was the home side that continued to create openings as the second period gathered pace.
Barcelona continued to struggle in the second period, with Neymar, Suarez and Messi all finding it difficult to produce their best form against an impressive Levante backline.
A second Barcelona goal on the counter was always a possibility as Levante broke forward, however, and that was indeed the case as Messi set up Suarez in the 92nd minute to ensure that the visitors collected all the points.