I had a very similar issue in 2014.
My A3, just 27 days out of its 3 year warranty and up to date with main dealer service history, decided that its EGR valve would fail. Main dealer said tough luck, it's £950 to fix take it or leave it.
Made contact with Audi head office, where I argued, successfully, that the part had failed prematurely and under consumer law, regardless of any warranty, they should take some responsibility for it.
They agreed, and admitted the part should be good for 10 years. they covered £750 of the bill leaving me to pay £200. They pointed out that I had been offered extended warranty and declined it, which was true and a fair comment.
Happy with and successful outcome.