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The Spanish champions confirmed their qualification following a hard-fought encounter that was settled in the final stages of the match.
After a prolonged struggle, Barcelona broke the deadlock 15 minutes from time through Danish defender Andreas Christensen, who scored in the 76th minute. English striker Marcus Rashford then sealed the result with a second goal in the closing moments of the match (90).
Elsewhere, the only major surprise came as second-division side Deportivo La Coruña eliminated La Liga team Real Mallorca with a narrow 1-0 win.
The remaining La Liga clubs in action on Tuesday also advanced to the next round. Elche defeated second-tier Eibar 1-0, Real Sociedad overcame third-division Eldense 2-1, and Valencia secured a 2-0 victory over second-division Sporting Gijón.
Attention now turns to Wednesday’s fixtures, with Real Madrid set to face third-tier Talavera, while city rivals Atletico Madrid take on amateur side Atletico Baleares.