Is it possible to fall asleep mindfully?

I've started to notice more and more often that my falling asleep is the exact opposite of mindful meditation; it's as if falling asleep is a "failed" meditation.

When trying to fall asleep, I sometimes try to remain mindful of my breathing. However, eventually my mind starts to wander, and sleep eventually comes when I get so absorbed in my thoughts, that my consciousness basically gets completely shut off, and only random, nonsensical thoughts remain (which basically turn into dreams).

Does anyone else share the same experience as I do? I don't experience how this is happening every night, but only when I'm drifting away in my thinking, and into sleep, and all of a sudden come back to awareness and notice what has happened.

So if meditation/super-consciousness is when there is pure awareness, how is it possible to fall into unconsciousness (sleep) mindfully? It seems as if my being mindful is what keeps me from sleeping in the first place.