Your instinct to pass a Connection to the persistence object is a good one, as is your reluctance to make the presentation tier responsible for acquiring it.
I'd recommend reading about the MVC pattern and looking into Spring. The Spring idiom will show you the proper way to layer an application. It'll help with your dependency issues as well.
UPDATE:
You'll have to read more about Spring.
Using Spring's common idiom of web->service->persistence interface layering will help layer your app properly.
Spring has dependency injection to help you manage your dependencies.
Spring's framework classes help you with acquiring and managing things like database connections.
I'm arguing that it makes little sense for you to rewrite what someone else has already written better. I'm suggesting that you start by browsing through Spring MVC Step By Step to see if you agree.
If you'd rather not learn Spring, I'd advise you to at least look at the classes they wrote for DataSource and JDBC connections. Maybe you can improve the way you're doing yours by looking at theirs.