Xavi's side had been woeful for the first 80 minutes and were deservedly 2-0 down, but Robert Lewandowski struck twice to claw Barcelona level and save his team from their first defeat of the season.
The fun didn't end there, however, as Joao Cancelo tapped home a late winner to seal all three points for Barcelona.
How the game unfolded
With 80% possession after the first ten minutes, Barcelona quickly learned the tone of this game. Celta were quick on the counter but were happy to sit deep with their back five to try and frustrate their hosts.
Marc-Andre ter Stegen had to backpedal to awkwardly dump Iago Aspas' attempted lob over the bar but Celta were celebrating soon after as Jorgen Strand Larsen beat the offside trap and found the German's bottom corner with real quality.
Jonathan Bamba bottled a glorious opportunity to add a second soon after. The midfielder found himself through on goal but his poor attempt to square the ball was both ill-advised and under-hit.
Barcelona's first real sight of goal came 23 minutes in. The in-form Joao Felix lost his composure as he drilled well over the bar from towards the end of the box.
That was as close as Xavi's men came in the first half and they were fortunate not to concede a second before the interval. Larsen's header drew an excellent save from Ter Stegen and Luca de la Torre could only blaze the rebound over the bar.
To try and fix things, Barcelona switched to a 3-2-4-1 setup at half-time but the hosts remained shockingly uncomfortable at the back and watched as Celta spurned a number of solid chances to double their advantage.
Out of nowhere, Ferran Torres had a glorious opportunity to equalise as he was played through on goal, but the winger dragged his effort wide of the target.
Barcelona were made to pay with 15 minutes to go as Celta profited from yet another counter. This time, it was Anastasios Douvikas who found the back of the net, scuffing an effort beyond Ter Stegen and offering some breathing room.
That lead would be cut down to one just minute later. Joao Felix's scoop found Lewandowski, who dinked the ball off the bar and into the back of the net to set up a nervy final ten minutes.
Lewandowski would be celebrating again just five minutes later, latching on to Cancelo's cut-back and rifling home to tie things up.
And Cancelo would complete the shocking turnaround soon after. The wing-back charged forward and was found by Gavi, steering his volley into the back of the net to seal the win.
Barcelona player ratings (4-3-3)
GK: Marc-Andre ter Stegen - 5/10 - The only reason Barcelona didn't concede four or five here.
RB: Joao Cancelo - 6/10 - Disappointing for the first 80 minutes but both scored and assisted in the final ten minutes to complete the turnaround.
CB: Jules Kounde - 4/10 - Not always convincing in possession. Made the wrong decisions at both ends of the field.
CB: Andreas Christensen - 3/10 - Sloppy. Had runners flying around him far too easily as he failed to track his men.
LB: Marcos Alonso - 2/10 - Similarly poor. His positioning allowed Celta to charge forward with ease on every counter. Didn't appear to retreat with any real urgency.
DM: Oriol Romeu - 7/10 - Did his best to pull the strings but also made a number of huge challenges at the back. Saved a certain goal denying De la Torre. Stunningly substituted at the break.
CM: Frenkie de Jong - 5/10 - Orchestrated the occasional positive movement before limping off injured in the first half.
CM: Ilkay Gundogan - 3/10 - A passenger for most of the game. Can't fault his work rate but Gundogan was just not involved at all.
RW: Ferran Torres - 3/10 - Not on it. Celta's deep back line proved far too compact for the winger. Missed an excellent chance to equalise in the second half.
ST: Robert Lewandowski - 7/10 - Like Gundogan, Lewandowski was clearly trying hard but was forced to deal with aimless crosses into the box as his teammates couldn't find a way into him. Took his two late goals really well.
LW: Joao Felix - 6/10 - Sloppy on the ball but was the only Barcelona forward who ever looked like scoring for most of the game. Should've done better with a handful of opportunities but it came as no surprise to see him set up Lewandowski's first.
Substitutes
SUB: Gavi (36' for De Jong) - 6/10 - A smart cross to find Cancelo and win the game.
SUB: Lamine Yamal (46' for Alonso) - 6/10 - Did his best to try and make something happen. Lively.
SUB: Ronald Araujo (46' for Romeu) - 6/10 - Stung the goalkeeper's gloves with a ferocious strike from range. Looks to be fit again.
SUB: Alejandro Balde (60' for Christensen) - 5/10 - Quiet in his time on the pitch.
SUB: Raphinha (72' for Torres) - 6/10 - Decent work to help turn the game around.
Subs not used: Inaki Pena (GK), Ander Astralaga (GK), Sergi Roberto, Inigo Martinez, Fermin Lopez
Manager
Xavi - 3/10 - Overthought his substitutions, switching formation and forcing players into uncomfortable roles. The decision to remove Romeu and leave a shaky defence without any natural cover was a major tactical oversight. Very fortunate to escape with a win.
