Did you check out this excellent blog on DI with Unity and ASP.NET MVC?
Should get you on the right track.
The answer is yes, you can - and the article shows you how.
In short, you create a HttpContextLifetimeManager to handle the "scoping" of your objects. The container "caches" the instance in the HTTP Context.
This is needed because the default life time managers provided by Unity don't cover HTTP Context scoping "off the shelf".
Of course, other DI container's (such as StructureMap - which i use), do.
Here is another (more up to date) article on the same thing, with the "Nerdinner" as the example.
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