Maxi Gomez's 67th minute goal put Celta ahead in a game they had to win in their fight to avoid relegation and Iago Aspas netted a penalty in the 88th minute to confirm Barca's third league defeat of the campaign.
With the league title assured last weekend, players such as Leo Messi, Luis Suarez, Ivan Rakitic, Gerard Pique, Jordi Alba and Sergio Busquets didn't even travel to Vigo as Barca boss Ernesto Valverde gave absolute priority to the Champions League.
Ousmane Dembele did start for Barca, but the French international winger lasted just six minutes before leaving the field with a hamstring injury. The club confirmed the injury, saying that Dembele would "undergo tests" on Sunday.
Earlier in the day Levante beat Rayo Vallecano 4-1 to virtually assure their salvation, condemning Rayo Vallecano to relegation.
Levante took advantage of some terrible defending from Rayo to go 2-0 up before halftime thanks to Jose Campana and Ruben Vezo.
Alvaro gave Rayo a chance with a goal 20 minutes from time, but as Rayo hurled players forward looking for an equalizer Levante sealed the win with late goals from Jason and Enis Bardhi.
Borja Iglesias scored twice as Espanyol defeated Atletico Madrid 3-0 to keep their hopes of playing in Europe next season alive.
A Diego Godin own goal opened the scoring for Espanyol in a game where Wu Lei played 79 minutes.
Real Sociedad still have a chance of playing in Europe next season after Manu Garcia's 24th minute own goal gave them a 1-0 win away to Alaves in the Basque derby.
The win takes Real Sociedad up to eighth in the table, while Alaves drop to tenth, although they have the same number of points.
Friday night saw Leganes ensure their survival with a 3-0 win away to Sevilla which also damages Sevilla's chances of finishing fourth.
Youssef En-Nesyri and Martin Braithwaite made it 2-0 in the first half and after withstanding Sevilla's attempts to draw level, Oscar scored a third goal with five minutes remaining.