How to properly handle irate customers?

The most obvious is not to become irate yourself.

Continue to be professional, take an actual interest in the cause of the customers irritation, identify if it is within your scope to resolve or does it need to be referred/escalated.  

If you promise a follow up, remember to make the follow up return call with as much information to resolve the issue.

Any irate customer when they recognise a genuine desire to help solve their problem often calm down.  

Remember to use a professional well balanced tone and smile - even if they can't see it they can feel it.