Robert Lewandowski missed a second-half penalty for Barcelona after Marcus Rashford had scored a left-footed volley in first-half stoppage time to give them hope of turning around a two-goal deficit.
Hansi Flick’s side were second best for long periods and deservedly fell to their first league defeat of the campaign against a side who had not beaten them in the Spanish top flight for just over a decade.
Former Barcelona winger Alexis Sanchez opened the scoring from the penalty spot after visiting captain Ronald Araujo had bundled Isaac Romero over by the byeline.