An apology is always necessary and will always be within your reach, regardless of your position in the company. This way, you can calm a stressed customer. Depending on the way you talk, you can achieve a great result.
You can exchange an “I’m sorry you feel that way” for “Please accept my apology for the mistake.”. As well, you can use empathy to demonstrate that you recognize the customer’s frustrations, such as “I understand how upset you are, I would feel the same way, and I apologize for what happened. Would you give me a chance to solve it?”.
By acting this way, you will give a voice to the customer, and they will feel in control, and if they accept this request, you can start the solution process. Otherwise, let them vent until they calm down.
Source: Mindtools, Dealing With Unhappy Customers