Her father is Dario Argento, an Italian film director, producer and screenwriter, well known for his work in the Italian giallo genre and for his influence on modern horror and slasher movies. Her mother is actress Daria Nicolodi and her maternal great-grandfather was composer Alfredo Casella.
When Argento was born in Rome, the city registry office refused to acknowledge "Asia" as an appropriate name, and instead officially inscribed her as "Aria" (a name accepted by the city registry). She nonetheless always went by the name Asia, which she later used professionally. Argento has said that as a child she was lonely and depressed, owing in part to her parents' work. Her father used to read her his horror scripts as bedtime stories. At age eight, Argento published a book of poems. At the age of 14, she ran away from home.
Argento has mentioned in interviews that she does not have a close relationship with her father. She has mentioned that he was absent when she was a child, and has also mentioned that, because of this, she did not have a happy childhood. Regarding her relationship with her father and her reason for acting, she has stated that:
I never acted out of ambition; I acted to gain my father's attention. It took a long time for him to notice me – I started when I was nine, and he only cast me when I was 16. And he only became my father when he was my director. I always thought it was sick to choose looking at yourself on a big screen as your job. There has to be something crooked in your mind to want to be loved by everybody. It's like being a prostitute, to share that intimacy with all those people.