The game had entered added time when midfielder Stephane Grichting equalised from close range following a corner.
The Boulogne players protested, arguing that the goal should have been disallowed because of a previous foul.
Auxerre's Belgium striker Luigi Pieroni ended Boulogne's amazing run with an angled shot from just outside the box on 98 minutes.
Coupe de France
Amateurs Nimes through to cup semis
Earlier, a goal in extra time gave amateurs Nimes a 4-3 home victory over Ligue 1 Sochaux.
Striker Matthias Verschave scored the winner on 104 minutes with a little help from Sochaux goalkeeper Teddy Richert, who failed to catch his seemingly harmless deflected shot.
Third division Nimes, who had knocked out three Ligue 1 clubs on their way to the quarter-finals, opened the scoring on 23 minutes, midfielder Ulrick Chavas finding the net after collecting a headed pass from fellow midfielder Stephane Beyrac.
Two blunders from Nimes goalkeeper Cedric Duchesne then led to as many goals from Togolese striker Mohamed Kader, who headed home after 27 minutes and tapped in from close range in the 39th.
The visitors thought they were home and dry when Brazilian striker Ilan put them 3-1 up with a low drive from inside the box on 57 minutes.
Nimes kept fighting, however, reducing the arrears courtesy of striker Eloge Enza Yamissi on 72 minutes before forcing extra time five minutes later with a header from a corner by substitute Moise Kande.
Monaco and second division Sedan advanced to the last four with victories over Clermont and Grenoble on Tuesday.
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