Assuming United States
As far as "Will they will allow you to do that?" Of course! (How do you think renters get utilities?)
As far as the responsibility to pay, that's up to you. If you are setting things up like that to prevent being swamped with bills which your girlfriend has no legal obligation to pay, that would work. However, if something were to happen between you, and she were to move out, then she'd probably cancel all of the utilities, and you'd have to re-acquire the service in your name, which may lead to service interruptions or early-termination fees on your girlfriend's side (I'm thinking about getting a multi-year contract on something like Internet/TV).
In my case, my wife's name is on our electric bill, our cable/internet bill, but I'm on the water/sewage and trash. It just happened like that because of who had free time at the moment to sign up for service. It's slightly annoying, because any time I call our providers, they won't allow me to make changes to the account without my wife present.