Not exactly grammar, but I'm in an argument. I claim that shops advertising "cheaper prices" are using incorrect English, as those shops are selling products - not prices.

I maintain the correct English would be "lower prices", or "cheaper products".

The opposing argument is that "common usage" overrides this, and therefore is now acceptable. Is there a definitive, argument-winning answer out there?

Your logic is sound, and I deem 'cheaper prices' to be wrong; however, the common usage argument is always a killer.

'Cheaper' should not change its meaning just because it's being used the qualify 'prices'.