Are you sure you don't mean "drift off?" This phrase is related to sleeping.
"fall asleep" simply means to begin sleeping.
"drift off" means that you sleep without meaning to; it's unintentional. For example:
"I drifted off during class today, so my teacher scolded me."
The person fell asleep without intending to during his class, so his teacher yelled at him.
Does this make sense?