Having overturned a 3-0 first-leg deficit at Real's Bernabeu stadium on Wednesday, Juve were cruelly knocked out by a soft penalty converted by Cristiano Ronaldo in stoppage time of the quarter-final second leg. English referee Michael Oliver pointed to the spot after Mehdi Benatia appeared to foul Lucas Vasquez.
The defender was also involved in an incident with Juventus manager Massimiliano Allegri before Cristiano Ronaldo slotted home the decisive goal that sent Madrid into the semi-finals at the Italians' expense.
"We should face and we did everything possible to go through", the Brazil global full-back said.
The manner of the loss, with Juve fighting back, coming so close to extra time and losing to a late Ronaldo penalty has triggered an emotional reaction among those associated with Madrid and Juventus.
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But Ronaldo is adamant that the decision was correct.
Oliver hesitated before pointing to the spot, and then came a mass protest by the Juventus players, which led to veteran goalkeeper Gianluigi Buffon receiving a straight red card.
Chillini said that it was the biggest robbery that he had faced in his life but suggested that it is not surprising since Real are always favoured by the referees.
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The early strike offered the visitors hope of repeating compatriots AS Roma's incredible second-leg turnaround against Barcelona the previous night and the Croatian struck again in the 37th minute, heading beyond Navas, who got a hand to the shot but could not prevent it going in.
The Spanish centre-back was initially seen up in the stands watching the game but soon made his way to the Real Madrid bench in the second half.
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120th goal in 150 Champions League appearances. The Andalucian's left the Allianz Arena with a goalless draw to go out 2-1 on aggregate courtesy of their first leg home defeat.
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Mundo Deportivo claimed Los Blancos have a "Masters in refereeing", insisting: "They are through to the semi-finals in their usual style: a controversial penalty".