Atletico won 2-0 at the Nou Camp after Julian Alvarez and Alexander Sorloth scored following Cubarsi's dismissal for a foul on Giuliano Simeone.
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Taking to social media, Cubarsi posted: "One action determines the match and the tie. That's football and I take responsibility for the result.
"There's still a long way to go in this two-legged tie and we're more united than ever: we're a family and we've always shown it."
He added: "Onward, with effort and determination, we will never give up."