The European champions were far from their best in coach Luis Enrique’s 100th game in charge, but David Navarro’s first-half own goal and Luis Suarez’s strike in stoppage time was enough to secure a 10th consecutive win in all competitions.
Barca now lead Atletico Madrid by three points and have a game in hand on Diego Simeone’s men.
Third-placed Real Madrid are now seven points adrift of Barca, but can cut the gap when they travel to Granada later on Sunday.
Barca’s 28-game unbeaten run matches the same feat of Pep Guardiola’s side in 2010/11 with Enrique’s men well on course to repeat their treble of Champions League, La Liga and Copa del Rey from last season.
“More than statistics, I am interested in the collective objectives that we have and we can only reach them at the end of the season,” said Enrique.
“We are very well-positioned in all competitions, but there is a long way to go.”
Barca needed a slice of luck to open the scoring on 21 minutes as Jordi Alba burst down the left and his ball into the box was deflected into his own net by Navarro.
Levante remain rooted to the foot of the table, but had won their previous two home games and deserved to be at least level at the break as they outplayed the Catalans in the latter stages of the first-half.
Verza fired a free-kick inches over before Jefferson Lerma had a great chance to level when he skewed horribly wide with just Claudio Bravo to beat.
Jose Luis Morales came closest to equalising when his low effort came back off the post eight minutes before the break.
Levante continued to enjoy the better of the game in the second period, but found clear-cut chances hard to come by as former Manchester United forward Giuseppe Rossi’s shot from outside the box was the only time Bravo was forced into action.
Barca had barely threatened themselves after half-time, but did seal the points with the last move of the match as Messi picked out an unmarked Suarez to fire home his 36th goal of the season.