Barcelona and Espanyol were both reduced to 10 men in the second half before Luis Suarez netted the winner for the home side at the Camp Nou.
Antoine Griezmann returned to Barcelona's starting line-up at La Ceramica and repaid the faith with a stunning goal, teed up by Lionel Messi.
Luis Suarez twice put Barcelona in front at Balaidos with his first goals since January but Iago Aspas bent in a late free-kick to earn Celta Vigo a draw.
Liverpool won the Premier League title after a span of 30 years as Chelsea defeated Manchester City 2-1 on Thursday.
Victory extends Barca's advantage at the top of La Liga to five points over Real Madrid, who can reduce the gap back to two when they restart at home to Eibar on Sunday.
Eden Hazard was looking beyond Real Madrid to the Euros after undergoing surgery on his right foot in March while some felt Luis Suarez had played his last game for Barcelona after having a knee operation in January.
Lionel Messi has been suffering from tightness in his right thigh and had not trained fully since last Tuesday.
Lionel Messi trained alone on Saturday at Camp Nou after sitting out the team's previous two sessions, raising doubts over his involvement in the trip to Mallorca.
Luis Suarez was injured for four months and is expected to return to action on June 13, when Barcelona take on Mallorca.
Patrice Evra said the nature of the threats meant he had to be protected by bodyguards.
Luis Suarez, recovering from right knee surgery, said it was "complicated" to talk about players joining Barcelona at this time of crisis with the coronavirus pandemic.
Barcelona's defeat to Valencia raised early doubts about Quique Setien's appointment and gave Real Madrid the chance to move three points clear at the top.
Valencia will provide Barcelona's new manager Quique Setien with his first true test and an opportunity to make his mark.
Barcelona issued a press release saying the surgery on Luis Suarez was for an "injury to the external meniscus of the right knee".
Luis Suarez and Lionel Messi are part of the potent Barcelona strike force that keeps them in the top slots of Spanish and European football.
A mistake by Barcelona's usually excellent goalkeeper Marc-Andre ter Stegen gifted Real Sociedad an equaliser in a thrilling La Liga contest.
Luis Suarez scored a goal with his backheel in a La Liga match against Mallorca and called it the "best goal of my career".
Argentina and Uruguay will face off on Monday evening in a match expected to pit Lionel Messi against his Barcelona strike partner Luis Suarez.