The 84th-minute goal secured a win that took Barca to 84 points with six games left, 13 clear of champions Real Madrid who earlier beat Real Betis 3-1 with a Mesut Ozil double.
A victory away to Athletic Bilbao next Saturday, and a Real defeat at city rivals Atletico Madrid would enable Barca to be crowned champions for a fourth time in five years.
In the late game, Valencia's push for a top-four finish was boosted with a 5-1 thrashing of Malaga at home when they scored four times in six first-half minutes.
Valencia climbed to fifth with 53 points, pushing Malaga down to sixth on 50. Betis are seventh with 48 points.
The top two fielded weakened sides with next week's Champions League semi-finals in mind.
Barca visit Bayern Munich for their first leg on Tuesday, and Real travel to Borussia Dortmund on Wednesday.
Barcelona lined up with four full-backs in defence, Eric Abidal making his first start in over a year since having a liver transplant and playing the full 90 minutes.
The French international partnered Adriano Correia in the centre of defence, while usual first-choice players such as Xavi, Sergio Busquets and Jordi Alba started on the bench.
World Player of the Year Lionel Messi failed to make the squad as he recovers from a hamstring strain.
Barca had an early chance to take the lead when Andres Iniesta tumbled to win a dubious penalty but David Villa's 17th-minute spot-kick was saved by Levante keeper Keylor Navas.
The Costa Rican went on to pull off a string of saves against a Barcelona side lacking spark and a sense of urgency.
Barca substitute Alexis Sanchez help break the deadlock six minutes from time, crossing from the right for Fabregas, who controlled the ball and fired low into the corner in a crowded area.
"Obviously it is difficult to think about any game other than the one against Bayern. It will be one of the games of the year," Spain international Fabregas told reporters.
"But it is good to be able to go into it with a win behind us."
MARCELO INJURED
In Madrid, Real were without regulars such as Xabi Alonso, Sergio Ramos and Sami Khedira and after a scrappy opening period were forced into an early change when Marcelo went down injured.
The Brazilian left-back was carried off on a stretcher with a muscle injury after falling awkwardly.
Betis, chasing a place in Europe next season, had an effort ruled out for offside early on and Dorlan Pabon skimmed the bar.
Cristiano Ronaldo carried Real's greatest attacking threat and soon after he had crashed a 25-metre shot against the upright, a moment of Ozil inspiration broke the deadlock.
The Germany midfielder cut in off the right flank, exchanged passes with Karim Benzema on the edge of the area, and lifted a finish inside the near post just before the break.
A breakaway started by O
