Trying to win an argument with your customer is just plain silly. When the goal is to win the argument, you present every information you have, demonstrating how well you know what you are talking about and unconsciously pointing out that your customer is clueless and should have kept her mouth shut.
But the customer is always right, you are the one that has to learn how to relate with them better. Most argument against a customer is usually on technical grounds, but we forget that this is not a technical problem. It is a mind game. They endorse or dethrone you by their purchase power. Never underestimate the amount of business a satisfied customer can bring, and never also underestimate the destruction, and bad blood a unsatisfied customer can send your way.
The most seasoned entrepreneurs I know never argue with their customers and clients, they merely listen to every line of the complain and promise to fix it with strong, re-assuring words. They then present their own cases from the point of view of the customer’s situation and perception. Such an approach rarely ever turns into a battle of words, and who is right and who is left.
Every business owner should definitely read up How to Win Friends and Influence People by Dale Carnegie and/or the modern day remix: Winning With People by John Maxwell.
