Jonas Eidevall's side were left stunned by a quickfire Mathilde Bourdieu brace in the second half, but Alessia Russo reduced the arrears before Jen Beattie forced extra time in the dying embers.
The Gunners had to come from behind in the additional 30 minutes as Julie Soyer's strike was cancelled out by Russo's second of the evening, but the ex-Manchester United striker then fluffed her lines from 12 yards - as did Frida Maanum - as Paris progressed to the second qualifying round.
The early gilt-edged chances fell Arsenal's way, as Chiamaka Nnadozie reacted well to tip Maanum's strike over the bar in the sixth minute, although she nearly went from hero to zero by coming for and failing to meet the resulting corner.
With Nnadozie out of the equation, Lotte Wubben-Moy won the header, but Paris defender Soyer was on hand to clear the England international's effort off the line.
The French side otherwise quelled the Arsenal threat with relative ease before the break, and Eidevall decided against calling upon either Beth Mead or Russo, who had been dropped from the first XI since the Gunners' success over Linkoping.
Maanum saw another chance fly wide of Nnadozie's goal in the 54th minute, and Arsenal's general passiveness in the final third would eventually come back to bite them, as Paris number nine Bourdieu stunned Eidevall's side with two goals in as many minutes.
Arsenal goalkeeper Manuela Zinsberger was seemingly expecting a Clara Mateo cross to sail out of play in the 55th minute, but the ball cannoned back into the danger area off the post, and Bourdieu headed into the arms of Zinsberger, who was behind the line as she caught the ball.
The Gunners barely had any time to compose themselves after the restart before Bourdieu doubled her and Paris' tally in the 57th minute, as Eidevall's side lost the ball inside their own half and Gaetane Thiney slipped Bourdieu through to finish through Zinsberger's legs.
The changes inevitably arrived for Arsenal following the 24-year-old's second, but Eidevall was indebted to Zinsberger for keeping his side in the tie in the 65th minute, as she thwarted Julie Dufour with an outstretched leg.
With their European hopes in severe jeopardy, Arsenal suddenly showed more intensity in the final third and cut the deficit in half 10 minutes before the end, as Teninssoun Sissoko lost the ball inside her own box under pressure from Russo, whose side-footed shot bobbled off Nnadozie and into the back of the net.
Paris initiated the backs-against-the-wall protocol as Arsenal huffed and puffed for an equaliser, but Maanum continued her profligate streak in front of goal with a header onto the roof of the net in the 82nd minute.
Eidevall's last throw of the dice saw Beattie introduced as a makeshift striker for a nine-minute injury time period, and the 32-year-old immediately vindicated her manager's decision with a dramatic leveller in the sixth minute of added time, peeling away from her marker to head home from Katie McCabe's delivery.
Arsenal remained in the ascendancy during the first 15 minutes of extra time, but their endeavours were not rewarded, and more sloppy defending allowed Paris to retake the lead within one minute of the restart.
A Louise Fleury cross from the left was allowed to drift across the box unchallenged, and the ball fell for 38-year-old Soyer to crash home the French side's third goal of the afternoon.
However, Arsenal once again made up for defensive shortcomings with effervescent play in the final third, and with three minutes remaining, Russo cut inside from the left and fired a peach of a 20-yard strike into the bottom corner to draw the Gunners level again.
Following five perfect penalties, Nnadozie saved from Maanum before Zinsberger denied Daphne Corboz, but Russo failed to bring Arsenal back on level terms as Nnadozie stuck out a strong hand to deny the England striker.
Faced with enormous pressure on her 19-year-old shoulders, Louna Ribadeira gave Zinsberger the eyes nonchalantly rolled the ball into the corner to dump Arsenal out of Europe, and the Gunners now have three weeks to dissect that defeat before opening their Women's Super League campaign against Liverpool at the Emirates on October 1.