The Spain playmaker's masterly performance at Levante's Ciutat de Valencia stadium helped maintain unbeaten Barca's three-point lead over Atletico Madrid, who earlier crushed nine-man Sevilla by the same scoreline.
The latest victory for Barca, their 12th in 13 matches, also put them 11 points clear of champions and arch-rivals Real Madrid, who slipped further back in third after they were upset 1-0 at Real Betis on Saturday.
An almost impeccable performance from Tito Vilanova's side, one of their best of the season, was crowned when Victor Valdes saved Jose Javier Barkero's late penalty and Barca have 37 points with Atletico on 34 and Real on 26.
"We just need to keep doing what we have been doing, this is the right path," Iniesta said in an interview with Spanish broadcaster Canal Plus.
"Atletico are our closest rivals right now and playing very well and they are candidates for the title," he added.
Malaga are fourth on 22 points after they thrashed Valencia 4-0 on Saturday, level with fifth-placed Betis and two ahead of Levante in sixth.
After Levante kept a dominant Barca at bay until half-time and created a couple of decent chances of their own, Iniesta sent Messi clear to open the scoring with a dink over Levante goalkeeper Gustavo Munua in the 47th minute.
Iniesta danced to the byline and pulled the ball back for Messi to net his and Barca's second five minutes later before he picked up Pedro's pass and crashed a shot into the goal off the underside of the crossbar in the 57th minute.
Fabregas's goal came six minutes later when Iniesta played him through into the area and he rounded Munua and finished from a tight angle, prompting chants of "champions, champions" from some of the Barca fans in the stands.
"We are only in November and the season goes through until May but we are very happy with the way things went today," sporting director Andoni Zubizarreta told Canal Plus.
"Iniesta did a lot of great things but I would say the team as a whole played extremely well," added the former Barca and Spain goalkeeper.
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Messi's 18th and 19th La Liga goals of the campaign put the Argentine World Player of the Year within three of the record for the most goals in a calendar year of 85 set by Germany and Bayern Munich striker Gerd Muller 40 years ago.
The prolific 25-year-old has scored 70 for his club and 12 for Argentina and is hot favourite to secure a fourth straight World Player award when the winner is announced in January.
The only negative for Barca was an injury to full-back Daniel Alves, who was forced off with what Zubizarreta said was a leg muscle strain in the 14th minute.
The Brazilian was replaced by Martin Montoya, meaning all 11 Barca players on the pitch at that point were products of the club's youth academy.
Atletico, who visit city rivals Real next weekend and Barca two weeks later, strolle
