Madrid - Coach Jupp Heynckes said Bayern Munich played their best match for years but were undone by a mistake from goalkeeper Sven Ulreich to allow Real Madrid to reach the Champions League final.
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"We are very disappointed," Heynckes said. "My team played an outstanding game. I haven't seen Bayern play as well as that for years.
"At this high level you can't make the sort of mistakes we made after half-time."
The unfortunate Ulreich gifted Karim Benzema his second goal of the match one minute after the break at the Santiago Bernabeu stadium, completely misjudging a back-pass from Corentin Tolisso to allow the French striker to slot the ball into an empty net.
James Rodriguez equalised for Bayern in the 63rd minute, setting up a desperate last half-hour in which the Germans pinned Real back but despite a handful of chances failed to score the goal that would have given them an away-goals victory on aggregate.
Real won 4-3 over the two legs and will play either Roma or Liverpool in the final in Kiev on May 26.