Giuly, 37, started on the wing for the hosts, who were playing at Lyon's Stade Gerland because their own Stade Ludovic Giuly was deemed too small.
Before half time Falcao, who only recently made his return from a thigh injury, was carried off on a stretcher and could now reportedly miss the World Cup in Brazil this summer if he has ligament damage.
Monaco continued to dominate in the second half and Falcao's replacement Emmanuel Riviere scored two further goals to ensure Monaco's place in the last 16 of the competition.
First he finished easily in the 63rd minute after a mistake from the Azergues goalkeeper before finding the corner with a swerving shot 10 minutes later.
It should have been four in the final moments but Yannick Ferreira-Carrasco somehow put the ball over an empty net after Riviere's unselfish cut-back.
The next round takes place on Tuesday, February 11 and Wednesday, February 12.