First off, let's get something perfectly clear. The customer is usually NEVER right. But you NEVER what to let them know that. It is much better to lose the argument and win the sale. That is more satisfaction them the other way around.
41 years of retailing and watching people lie, steal, cheat, switch tags, return items they never bought from me and suffering about every kind of insult and abuse, short of violence, and I have learned that lesson well.
But in this case, I think you already know the answer. If you do not have it in writing, you have a weak case. There is such a thing as a verbal contract, but you would have difficult time enforcing it sans a legal action. Depending on how much money you are talking about and how strongly you fell about it maybe consult a lawyer or consider a small claims court action.