With a 3-0 first-leg advantage, Real looked firm favourites heading into Tuesday's game but Angel di Maria's missed penalty and a first-half brace from Marco Reus set Madrid nerves jangling.
They survived more Dortmund pressure in the second period, Henrikh Mkhitaryan striking a post while Iker Casillas was also called upon to make some crucial saves.
And coach Ancelotti confessed afterwards that his charges had almost paid for their poor first-half showing.
"All games at this stage are very difficult," he said. "We are fortunate to be through to the semi-finals.
"We're happy to reach the semis, but not happy as we suffered a lot. Especially in the first half. We opened the tie up again.
"We were better in the second half. Football is like that, we gave away gifts, lots of errors, no confidence, and we suffered because of it.
"Casillas made fantastic saves in important moments of the game. His game was very, very good."
Casillas himself pointed to basic errors before the break, with Pepe and then Asier Illarramendi surrendering possession in the build-up to Reus' two goals.
"Maybe it's good for us to receive a wake-up call like that from time to time," he said. "It's better that it came tonight and not in the upcoming games we have.
"The first goal we conceded was from our mistake and the second came from another error in midfield.
"We also missed a penalty. The good thing was that we then knew how to deal with being under pressure."
