Don't worry. I made you a promise. I won't go back on you.
I'm familiar with "go back on your word". What does "go back on you" mean here? Does it have the same meaning (keep my promise), or is it more like I won't disappoint you?
Yes, exactly. The context specifies that meaning.
That's also the most probable meaning without context, but it might have other regional connotations that I don't about.