Arsenal, who won the competition in 2007 and reached the semi-finals last year, scored late goals through Alessia Russo and Jen Beattie to cancel out Mathilde Bourdieu's quickfire double and take the match into extra time.
After Louise Fleury scored in the 106th minute, Russo bagged her second to force a shootout, but Arsenal's new signing had her spot-kick saved by Chiamaka Nnadozie as the English side lost 4-2 on penalties.
Arsenal, who finished third in the Women's Super League table last season, begin their domestic campaign at home to Liverpool on October 1st.