At club level, Asilian started swimming at Greek powerhouse Olympiacos youth Academy at the age of 6.[3] Olympiacos women's water polo department was founded in 1988 and Asilian went on to play for Olympiacos for a total of 29 years (including her youth career) before her retirement at the end of 2006–2007 season at age 35.[3][4] Asilian is the record goalscorer in the history of the Greek Women's Water Polo League with 703 goals, a record she achieved in a 15-year span (1992–2007).[5] She won 2 Greek Championships (1994–95, 1997–98) and the fourth place in the 1995–96 LEN European Cup with Olympiacos.[6][7]