A 2-2 draw in the second leg at the Santiago Bernabeu on Tuesday saw Madrid reach a third straight final, winning 4-3 on aggregate.
But a shocking error by Sven Ulreich proved costly for Bayern, the goalkeeper misjudging a back pass and gifting Karim Benzema his second goal of the game.
Hummels lamented the way Bayern gave away another goal, saying his side looked the better of the two teams in the tie.
"We were overall superior and had chances without finishing, but we have given Real a goal in both games and that is costly at this level," he told Sky.
"It brings us nothing …. we have [put up] a great fight, but in the end [we are left] with nothing."
Joshua Kimmich again opened the scoring for Bayern before Benzema's brace, while James Rodriguez dragged the visitors back into the tie against his parent club in the second half.
But Madrid were able to hold on, although Bayern defender Niklas Sule said his team deserved more.
"We had many opportunities. In my opinion, we had to get two penalties," he said.
"It's very annoying that we're out, especially since we were the better team in both games. We are all very disappointed."