The stunning victory has moved the Catalan giants three points clear of second-placed Atletico Madrid, still with a game in hand, which is away to Sporting Gijon next week.
Celta were missing a number of key players, including Nolito, Iago Aspas and Fabian Orellana, but put in a strong performance in the first period, before being undone by three Luis Suarez goals in the second period - the third of which came from a Lionel Messi assist from the penalty spot.
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Barcelona's first clear opportunity came in the 12th minute when Messi attempted to find the corner with a curling free kick, but the Argentine's effort just swung wide of the post, before Sergio Alvarez kept out Neymar from close range.
Messi then combined with Neymar in the 19th minute as Barcelona came close once again, but the former was crowded out at the vital moment and Alvarez claimed with ease.
Celta continued to impress and had a golden chance to take the lead in the 25th minute. A run from Daniel Wass eventually saw the midfielder slide a pass in to Carles Planas, who might have found John Guidetti in space, but the full-back's powerful strike was saved by Bravo.
It was Barcelona that took the lead in the 28th minute, however, when Messi curled a stunning free kick into the top corner to register his 13th league goal of the season.
Claudio Beauvue tried the spectacular for Celta five minutes later, but his volley, which was roundly clapped by the home supporters, was kept out by Bravo.
The visitors had their equaliser six minutes before the end of the first period and it came from the penalty spot after Jordi Alba had clipped the heels of Guidetti inside the box. The referee did not need long to make up his mind and Guidetti stepped up to send his spot kick into the bottom corner and leave the two teams on terms.
Sensational feet from Neymar set up a chance for Suarez barely two minutes into the second period, but the Uruguay international's effort hit the post, before Sergi Roberto fired wide.
Barcelona continued to press for a second and had another excellent chance in the 56th minute, but Alvarez was on hand to deny Andres Iniesta.
Alvarez could do nothing but watch in the 59th minute, however, when Messi set up Suarez, whose half-volley nestled in the back of the net.
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Guidetti remained a threat for Celta and might have levelled 10 minutes later when he forced Javier Mascherano into a defensive mistake, but the Swede could only fire wide from inside the box, before Wass also missed the target.
Wass had another stunning chance in the 74th minute, but he fired straight at Bravo from inside the box. That miss proved all the more costly one minute later when Barcelona went down the other end and scored a third. Messi found Neymar, who rounded the goalkeeper and crossed for Suarez, who poked the ball home for his second of the match.
Suarez then had his hat-trick in the 82nd minute when Messi, from the penalty spot, passed to his forward teammate and the 29-year-old smashed home from close range. Rakitic then got in on the act for 5-1 when he lobbed Alvarez after collecting a pass from Suarez.
There was time for one more, however, and it came for Neymar, who took a Suarez pass out of the sky before chipping Celta's goalkeeper from close range in the first minute of stoppage time.