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The hosts had defender Tuta sent off late on but Barca could not capitalise.
That dismissal for two bookings capped a busy evening for English referee Michael Oliver, who earlier awarded Frankfurt a penalty before changing his mind after being sent to the pitchside monitor.
Former Manchester City attacker Torres’s goal means the Spanish side, now unbeaten in 22 games and rejuvenated under manager Xavi, will be favourites to advance to the semi-finals when they meet again for the second leg at the Nou Camp next Thursday.
Although Barcelona had more of the ball and finished the game on the front foot, Eintracht Frankfurt were arguably the better team on the night. The Bundesliga club had 18 shots (compared to Barcelona’s seven), despite having just 34% of the possession.
Pierre-Emerick Aubameyang and Adama Traore, who started in place of Ousmane Dembele, had poor outings. Meanwhile, Gerard Pique was taken off early in the game with what appeared to be a groin injury.
For Frankfurt, Martin Hinteregger turned in a colossal display at the heart of their defense. Filip Kostic and Rafael Borre were a constant threat to Barcelona’s backline.
The winner will meet either West Ham or Lyon in the semi-finals, with that tie evenly poised after a 1-1 first-leg draw at London Stadium.