A devastating second-half spell yielded four goals in just eight minutes for the defending Liga champions, who were beaten at Camp Nou by the Basques.
Two goals in the last five minutes of the first half propelled the Red Devils three points and have plunged the champions into even greater trouble. Manchester United gave their top-four aspirations a major boost as they cruised to a 3-0 win at Premier League champions Leicester City on Sunday.
Marco Silva's men gave their survival hopes a huge boost with victory at the KCOM Stadium as the Reds' winless run continues.
The South American duo did the damage in the first half with two fine goals, before Antoine Griezmann gave his side a glimmer of hope after the break.
The minnows hit the woodwork twice before Alex Alegria struck, but Luis Enrique's men saw Jordi Alba's goal moments later wrongly disallowed.