Tiago put the hosts ahead in the 14th minute, making the most of some poor goalkeeping by Iker Casillas, before Saul - a replacement for the injured Koke - doubled their lead with a stunning overhead kick.
Atletico blew the Liga leaders away in the second half, with Antoine Griezmann and Mario Mandzukic completing a resounding triumph to move the champions to within four points of Real.
But Carvajal was scathing of his side's efforts early in the contest post-match.
He told reporters: "That could be the worst 30 minutes Real Madrid have ever played, gifting chances and creating nothing ourselves.
"We're still angry, but the truth is in the result and we're definitely feeling pretty down.
"I would like to ask our fans for forgiveness. We were not up to the task today and we have to congratulate [Atletico] because they were the better team.
"What we have to do is forget this match and this result as quickly as possible.
"It's true they have won the last few derbies, but let's wait until the end of the season and see who has the titles."
