Hola Colin

The equivalent of "to apologise" is "pedir perdón" -we do paraphrase it. This is quite commonly used. There is also a more "direct" way to say "to apologise", which is with the verb "disculparse", as a reflexive verb. So you can also say (although it sounds a bit more formal):

Tú te disculpaste.

The problem with disculparse is that you don't use it when you're saying you apologise "to someone" (which is what we're doing in that sentence. So in this case, the only way to use a verb that completes the meaning is "pedir perdón [a alguien]", so you can include who you apologise to: Me pediste perdón.

I hope it clarifies it.

Saludos