I'm sorry that this happened. And i'm sure that if i were in your shoes, I want to justify and rectify things for my child too. BUT..I know that deep down, the RIGHT thing for me to do, really...is to teach my daughter that it's how she deals with things in life that really matters, and NOT expect how others should change themselves. Of course, we should have expectations of others...but we have to have a healthy mindset in knowing that people can only change themselves. I want my daughter to know as early as possible that we will all go through life having things happen that are very unfair too us. We only have control over reaction.
It's more dangerous if our children grow up thinking that everything has to be fair because if they don't know how to deal with mistakes, mishaps, even down right cheaters...they will be seeking justice for every little thing. That everything is owed to them...It can lead to a very dangerous way of thinking.
I would encourage your daughter that what really matters is that she did the right thing and she knows that. And that her family believes her and in the end, they are the only people that truly matter. Sometimes honesty isn't always rewarded. But that makes for a very noble person...Someone who does the right thing, even when no one is watching...NOT the people who do it to get praises.
Tell her that you are proud of her (I'm sure you do) and that in the end all it matters is what her mommy and daddy thinks of her. An, of course, you think the world of her!
It doesn't mean that you shouldn't talk with the teacher if you can. Because she should at least know what's going on with her students. That's really something a parent should be able to talk to the teacher about. But I wouldn't get your daughter involve in the conversation...she shouldn't have to know.