The Catalan giants had won 12 out of 12 on home soil heading into this match and required victory today to make it a full year without dropping any points.
Maxi Gomez levelled up after Barca had turned the game on its head with goals either side of half time, meaning that the league leaders have now drawn their last two matches to open up the La Liga title race.
With 24 goals being scored in the last four meetings between these two sides, a lively contest was expected and it more than lived up to the hype.
Having been dealt a blow just a few minutes into the match with the loss of defender Sergio Gomez, Celta had to sit back a little as Lionel Messi and Luis Suarez both got shots away, the former at the end of a trademark slalom run.
The Sky Blues set up positively in terms of formation, with seven-goal Iago Aspas and Gomez playing alongside each other in attack, and they were rewarded 20 minutes in when their top scorer found the net.
Barca were left completely exposed at the back when Samuel Umtiti was drawn high up the pitch and, after Marc-Andre ter Stegen produced one of the saves of the season to block the initial shot, Aspas was there to tuck home the rebound.
Less than two minutes later the Catalans were level, thanks to a smart finish from Messi to tuck the ball through Ruben Blanco's legs after being played through by Paulinho - his first goal since October 28.
Suarez then put his side ahead moments later when latching on to an impressive Messi through-ball, only for the officials to wrongly flag him for offside.
Further chances fell the way of Paulinho and Daniel Wass in a gripping conclusion to the first half, neither of whom were able to keep their composure from close range.
The first big chance of the second period fell the way of Paulinho, who did the hard work by rounding the opposition keeper before finding the side-netting from a tight angle.
A subdued Barca were temporarily brought back to life when Denis Suarez was introduced from the bench to provide some midfield creativity, with a fine passing move culminating in the hosts' second.
Suarez, with just two goals in his last eight games prior to this one, was picked out by Jordi Alba for a simple finish to put the league leaders ahead for the first time.
That lead lasted less than 10 minutes as, after a couple of Blanco saves, Aspas left Umtiti for dead down the right-hand side - the defender pulling up injured in the process with a hamstring strain - and then had the quality to spot attacking teammate Gomez to do the rest from 10 yards.
Paulinho fired over the bar from a good position and Gerard Pique clipped the outside of the post as Barcelona quickly responded, while Messi also nodded inches wide.
Celta were simply refusing to sit back, however, as Pione Sisto tested Ter Stegen in the closing stages, though the final chance of the match fell the way of substitute Paco Alcacer, who failed to keep his header on target when under little pressure.
Barcelona: Ter Stegen; Sergi Roberto, Pique, Umtiti (Vermaelen 72'), Jordi Alba; Busquets, Iniesta (D.Suarez 53'), Rakitic (Alcacer 79'), Paulinho; Messi, L.Suarez
Celta Vigo: Ruben Blanco; Mallo, Fontas, S.Gomez (Cabral 4'), Mendez (Roncaglia 63'); Jozabed (Hernandez 84'), Lobotka; Wass, Sisto, M.Gomez; Iago Aspas