Mathilde Bourdieu scored twice in three minutes for Paris early in the second half to leave Arsenal in an uphill battle, but substitute forward Alessia Russo clawed a goal back on 80 minutes before defender Jen Beattie levelled the scores deep into added time to force extra-time.
Replacement Paris forward Louise Fleury netted in extra-time, again putting Arsenal on the backfoot before Russo struck once more on 116 minutes.
However, Paris held their nerve in the shootout, claiming a 4-2 victory on penalties and a place in the second qualifying round.
Arsenal were without stars Leah Williamson and Vivianne Miedema, who remain out with knee injuries, but still featured a strong side that included Katie McCabe, Kim Little, Caitlin Foord and Lotte Wubben-Moy.
England forward Beth Mead was an unused substitute as she continues to recover from an ACL injury that forced her out of the Women’s World Cup.