In Breda, Switzerland took the lead on 19 minutes when Eve Perisset was sent off for bringing down Ramona Bachmann.
In Breda, Switzerland took the lead on 19 minutes when Eve Perisset was sent off for bringing down Ramona Bachmann and Ana-Maria Crnogorcevic slammed in a header from the ensuing free-kick.
In the 76th minute, Abily's free-kick somehow slipped through the fingers of Swiss goalkeeper Gaelle Thalmann, following Crnogorcevic's foul on Eugenie Le Sommer.
In Rotterdam, Austria cruised to an easy win against Iceland to seal top spot in Group C with first-half goals by Sarah Zadrazil and Nina Burger, and a late strike by substitute Stefanie Enzinger.
Zadrazil fired home on 36 minutes, capitalising on a howler by keeper Gudbjorg Gunnarsdottir who failed to deal with an easy cross.
Burger nodded in Zadrazil's flicked header eight minutes later, and Enzinger volleyed home on 90 minutes to round off the scoring.
France started on a high, with Claire Lavogez's blast saved by a diving Thalmann.
But things got complicated as Perisset left the pitch in tears, with France struggling against the solid Swiss defence.
Thalmann kept out Abily's low shot, while Swiss skipper Lara Dickenmann saw an effort repelled by Sarah Bouhaddi.
After the equaliser, France held off Switzerland's pressure, with Crnogorcevic firing wide and Bachmann shooting straight at Bouhaddi.
"Believe me, it's a great satisfaction," said French coach Olivier Echouafni.
"We believed in it right to the end and our equaliser, considering we had ten against eleven for so long, was thoroughly deserved."
"It was quite a night and it was quite unexpected. We didn't panic, though. At half-time, I knew we'd get a chance with a free-kick, and so it was."
Swiss coach Martina Voss-Tecklenburg said the first half had gone as planned.
"We showed a lot of running, willingness, determination and tactically we were great."
"It was a real pity -- we fought so hard, and then we concede such a goal. We were so, so close to beating France. I can just hope my team learn from this."
Austria never left any doubt about who was going to dominate the game in Rotterdam, with Nicole Billa testing Gunnarsdottir three times in the first half.
In a rare Iceland chance, Fanndis Fridriksdottir narrowly missed the goal on 26 minutes.
The Austrians kept Gunnarsdottir busy until the end, as the Iceland keeper had to handle second-half attempts by Laura Feiersinger, Zadrazil and Sarah Puntigam.
"Today we were very quiet and controlled the game in most phases," said Austrian coach Dominik Thalhammer.
"It's a fantastic day for Austrian women's football and Austrian football as a whole."
"We are now in the quarter-finals. Of course we want more and anything can happen in one game."
Iceland coach Freyr Alexandersson said he was disappointed with his players who "played very badly."
"I knew this match would be very difficult for us if we get a goal against us and then we got this horrible goal," he said.
"I'm disappointed not with this tournament as a whole but with this match," he added, praising the Iceland fans for "the support we had in Holland and also at home."