Twice women's Champions League winners Lyon won 6-5 on penalties following a 1-1 draw after the referee had ruled out Montpellier's final attempt in the shootout.
However, TV replays showed the ball had crossed the line after hitting the post and bouncing against the keeper's arm.
"Olympique Lyon confirm they have offered Montpellier a replay of their French Cup semi-final... [for reasons of] fair play and taking into account the good relationship between the clubs," they said in a statement.
"They have done it like Arsenal and their coach Arsene Wenger did in 1999," Lyon added, referring to an FA Cup Fifth Round game against Sheffield United at Highbury.
Wenger offered to replay a game Arsenal won 2-1 having scored the second goal after the ball was put into touch by United so one of their players could be treated for an injury.
Arsenal's Nigeria forward Kanu, unaware a United player needed attention, received the ball from the resulting throw-in, broke down the right and squared for Marc Overmars to score.
Wenger was later presented with a fair play award by European football's governing body UEFA for his gesture.
Lyon's women have already clinched their seventh consecutive French league title this season with maximum points. They defend their European crown against Germany's Wolfsburg on May 23.
‘The pure simplicity of the way Slot has managed the squad is probably the biggest thing I could say about him. It’s not broken, so let’s get on with it’: Liverpool legend full of admiration for Jurgen Klopp's successor at Anfield
‘I trained at Spurs and thought they’d give me a chance. But I received a letter thanking me but saying they didn’t think I was good enough – I was gutted’: How Tottenham missed out on signing Wales legend John Toshack
