Strozzi was the daughter of Isabella Griegha and an unknown father. Her mother was a servant of Giulio Strozzi, a well-known citizen of Venice, and it is generally thought that he was her birth father. In any case, he adopted her, named her his heiress, and had her educated very well, which was unusual for the time.
At 16 she was already a respected musician and began composing as well, setting words by her father to music. In 1644 she published the first of eight collections of music. She became famous not only for her music but for her position as leader of the salon surrounding her father; a not entirely respectable reputation, and she had four children by unknown fathers, whom she raised herself. She is the only prominent woman composer of her period and region.