BVB striker Robert Lewandowski netted four times to help the Bundesliga side to a crucial home win in the first leg of the semi-final tie, prompting Jose Mourinho to blame Madrid's defence for the loss at Signal Iduna Park.
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"It's more a team failure than an individual one. Lewandowski played really well and we failed to stop him," Varane told Uefa.com.
"We didn't expect a result like this. We expected a much more positive result.
"We approached the game with plenty of ambition, but we went down to a very aggressive team who were compact and gave us no space."
Xabi Alonso, meanwhile, admitted that Madrid were outplayed by Jurgen Klopp's men, but believes they can still turn things around.
"Borussia competed better than us and we have to learn for the next match because we have little margin for error," the midfielder told reporters.
"We can do it in the second match at home, even though we are not happy with this match right now. We have six days to wait and we can turn this result around."
The result at the Santiago Bernabeu takes place at April 30.