Monaco will travel to Bordeaux in the next round. The French champions dispatched second division Ajaccio 5-1 on Saturday.
Lyon dominated the first half and found the back of the net just before the interval when defender Jean-Alain Boumsong headed home from an Ederson corner.
Ligue 1 top scorer Nene levelled the scores from the penalty spot early in the second half after the Brazilian fell in the box although TV replays seemed to show that Sidney Govou had not touched the Monaco striker.
The principality side kept on pushing and South Korean forward Park Chu-young headed the winner from a Francois Modesto cross after Lyon keeper Hugo Lloris misjudged the ball with less than 15 minutes left.
"We played against a great team. They led at the interval... but we put a little bit of madness and conviction in our game after the break and that is what is required when you play a Cup game," Monaco coach Guy Lacombe told France 3 TV channel.
"We'll have to play at an even higher level in the next round as we play at Bordeaux. We'll do what we can," he added.
Paris Saint-Germain ended a three-game losing streak in all competitions when they beat third division Evian 3-1 at home.
Turkey forward Mevlut Erding put the capital side in front through two headers either side of the interval. Evian narrowed the gap to one goal through Kevin Berigaud but striker Guillaume Hoarau sealed PSG's victory in stoppage time.
Paris will visit fourth-division Vesoul in the next round.
Holders Guingamp, who are 17th in Ligue 2, reached the round of 16 with a 1-0 win at fourth division Mulhouse.
Ligue 1 strugglers St Etienne and Nancy went through to the last 32 after the two remaining rescheduled round-of-64 games went ahead on Sunday.
Dimitri Payet scored twice to help St Etienne thrash Lorient 4-1 at home in an all-Ligue 1 tie.
Nancy needed a penalty shootout to overcome fifth-division Thiers after the teams drew 1-1 after extra time.
St Etienne will play fourth division Villefranche in the round of 32 while Nancy host third division Plabennec.
