Aghdashloo was born in Tehran, Iran, as Shohreh Vaziri-Tabar, the daughter of Effie (née al-Sadat) and Anushiravan Vaziri-Tabar. She has three brothers: Shahram, Shahriar, and Shahrokh. Her stage name is from the family name of her first husband, painter Aydin Aghdashloo. After their marriage in 1972 when she was 19 and he was 31, she began attending theatre workshops, against the wishes of her family. She had always wanted to be an actress, and soon began playing leading roles in Iranian theatre and film. They did not have children and were divorced in 1979, when she left Iran for England at the start of the Iranian Revolution.
Once she arrived in England, she earned a Bachelor's degree in International Relations because of her interest in politics after having to leave her home country. She was already familiar with England, as her parents took her to London as a child. She then continued to pursue her acting career, which brought her to Los Angeles. In 1987, Aghdashloo married actor/playwright Houshang Touzie. They have a daughter, Tara Touzie, born in 1989. She has since performed in a number of Touzie's plays, successfully taking them to national and international stages, primarily in the Iranian community.