Iago Aspas' free kick in the 88th minute earned Celta a 2-2 draw after Suarez had given Barcelona a two-time lead with assists from friend Lionel Messi in both cases.
The result means that Barcelona have beaten Real Madrid by one point at the top of the standings, but Los Blancos have played one less game and have the advantage if the teams finish tied for points thanks to a better head-to-head record. -head.
"We have little room for error now and in theory we have to win all of our games and hope that our rival slips," said Barcelona manager Quique Setien, quoted by Marca. "We will try to continue to collect the maximum points."
Fighting from Barcelona to Vigo has continued, having won only once in the league since 2015.
Vigo was perhaps 16th but had conceded only once in his previous seven games. While Barcelona were able to violate the defense of Celta Vigo, the opposing team could not defend.
Suarez gave Barcelona a 20-minute lead, catching Vigo off guard.
Messi, taking a free kick just outside the penalty area, passed to Suarez instead of himself shooting and the Uruguyan made no mistake about his first goal since returning from an operation at knee.
But at the start of the second half, Vigo stabilized on the counter.
With Barcelona's defense out of position in the 50th minute, Vigo advanced on a break from 2 to 1. Fyodor Smolov ended the movement with a simple finish.
Suarez's clinical arrival inside the area in the 67th minute restored Barcelona's advantage before Aspas' low free kick around the wall completely caught Barcelona goalkeeper Marc-André ter Stegen.
"We shouldn't have gotten into this situation because we were the best team in the first period," said Setien. "We must have entered at break with a more comfortable score."
Barcelona are now hoping that their local rival, Espanyol, can help them on Sunday when they host Real Madrid. But Espanyol occupies the last place in the ranking.
End of the Bundesliga
Meanwhile, on the final day of the Bundesliga in Germany, Bayern Munich - which won the title earlier this month - was able to hoist the league trophy for the eighth consecutive time.
As in Vigo, there were no fans because the matches were played behind closed doors due to the coronavirus pandemic.
There have been no changes to the four places in the Champions League, although Borussia Dortmund suffered their worst home loss since 2009 after a 4-0 loss to Hoffenheim.
At the bottom of the standings, Werder Bremen avoided relegation by beating FC Cologne 6-1 at the expense of Fortuna Dusseldorf, which lost 3-0 to Union Berlin. Bremen will now play in the relegation qualifier while Düsseldorf will advance to the Bundesliga 2 next season.
In England, Manchester United became the first team to reach the FA Cup semifinals by beating Norwich 2-1 in overtime. The other three quarterfinals will take place on Sunday.
