It's all in the way you drive your car and of course newer cars may be more economical than those that are older.
I have always driven automatics and some of them were more economical than others depending on their engine size and age but one thing which I found is important is to drive it gently, I don't mean move so slowly that you annoy everyone else on the road but just don't thrash the engine!
These days I don't think automatic cars are an awful lot different to manual cars. My partner drives a manual car and knows a lot about engines and miles to the gallon. He thinks my Renault Clio 1.4 is really economical compared to the Hyundai Amica 1.0 I used to own. He puts it down to the type of automatic transmission that a car has. There are various types and some are more economical than others.