Real headed into the tie 2-1 down on aggregate but looked set to book their place in the final after racing to a two-goal lead inside 15 minutes thanks to a brace from Cristiano Ronaldo.
However, Arjen Robben soon levelled matters for Bayern from the penalty spot and with neither side being able to find another goal, the match went to penalties - where Real proceeded to lose 3-1 after misses from Ronaldo, Kaka and Sergio Ramos.
And Casillas, who did his best to help Los Blancos win the tie with saves from Toni Kroos and Philipp Lahm in the shootout, admits the defeat was hard to take, although he now wants his players to focus on their bid to win La Liga.
"It's a big blow for us," the Spain international said after the game. "It was a very exciting game, but penalties are always a lottery.
"We played well, but Bayern are a great team. We now have to concentrate on the league."
Real midfielder Xabi Alonso echoed his captain's sentiments and admits the players are coming to terms with the premature end to another continental campaign.
"Of course we are quite disappointed, but we are coping," Alonso said. "It's part of football.
"Today [Wednesday] we are experiencing the other side of football but we knew that could happen.
"We wish Bayern the best of luck and congratulations to them."
Real currently hold seven-point lead in the league and will look to move one step closer to the title on Sunday when they entertain Sevilla.