I'm not sure what you mean by "asleep", but on my iPhone 3GS all I have to do is start the video and then press the power button (on the 3GS) and the video will continue to stream to the Apple TV over Airplay. When I do this the screen on the 3GS stays off until I wake up the phone at which time I see the lock screen. This also works with 3rd party video streaming apps and as far as I know it's been that way for a long time -- even under iOS 5 (also works on iOS 6, which I just installed yesterday).
My iPad behaves the same way and neither my iPad nor iPhone is jailbroken.