West Ham striker Daniel Uchechi was found unmarked by a cross from the right to put Nigeria ahead with a far post header six minutes after the break.
Kopplin leveled the score just one minute later with a smart finish from inside the area.
Germany's hopes were dealt a blow on 64 minutes when Tobias Kempe received a straight red card for stamping on the knee of Nurudeen Orelesi.
Nigeria made their numerical advantage count four minutes later after Rabiu Ibrahim scored his first goal of the tournament to restore the Africans' advantage.
Substitute Mario Vrancic restored parity once again in the 75th minute with a close-range shot after being set-up by Kopplin, who stole the limelight in stoppage time.
A brilliant solo effort saw him run half the length of the pitch, round a defender and send a low shot into the bottom corner.
Germany will face Brazil, who beat Uruguay 3-1, on Saturday for a place in the semi final.