The La Liga leaders beat their German rivals 2-1 at the Santiago Bernabeu in the second leg on Wednesday after losing the away game with the same score, but Bayern eventually booked their ticket for the final when they overcame Madrid 3-1 in the penalty shoot-out.
"The team are having an exceptional season. Our numbers in La Liga are incredible and we've gone very far in the Champions League, which is expected of us. We cannot ask our players to do more," Butragueno was quoted as saying on the Madrid website.
"The team tried everything to reach the final and we're proud of our players. We faced a team with great character that managed to recover quickly from conceding two goals in the first half. We also improved but were unable to break their defensive line.
"Exhaustion was noticeable on both sides. Penalty shoot-outs are unpredictable and often won by the most inspired side."
Real Madrid return to La Liga action on Sunday when they welcome Sevilla to the Bernabeu.
How do you stay up to date with football when on the move? With http://m.goal.com – your best source for mobile coverage of the beautiful game.