The Colombia international appeared to damage his knee following a strong tackle from behind, prompting the introduction of Emmanuel Riviere, who scored twice
The former Atletico Madrid striker opened the scoring at the Gerland just before the half-hour mark only to then be stretchered off with a suspected knee injury on 41 minutes.
The loss of Falcao for a significant period of time would represent a massive blow for Monaco's Ligue 1 hopes but it mattered not in this Coupe de France clash, with the visitors running out comfortable winners thanks to second-half strikes from Falcao's replacement, Emmanuel Riviere.
Much of the build-up focused on former Monaco star Ludovic Giuly, who made good on his promise to return to hometown club Chasselay last year.
However, Falcao quickly came to the fore, snatching at a great chance to give Monaco an 11th-minute advantage.
The Colombia international did not have to wait long, though, to break the deadlock, Falcao tapping home unchallenged after Valere Germain had headed Joao Moutinho's free kick across the six-yard box.
However, just four minutes before the break, Falcao's joy turned to anguish, after he was left clutching his left knee in agony following a rough challenge from behind from Soner Ertek.
Monaco were stunned by the loss of their star summer signing but in Riviere they had a more than worthy replacement.
After Yannick Ferreira Carrasco had hit the post, the 23-year-old doubled Monaco's advantage in the 63rd minute when he hooked the ball home after latching onto a long ball over the top.
Riviere struck again soon after, this time producing a fine finish after terrific work from Ferreira Carrasco, who subsequently blew a golden opportunity to make it 4-0 by ballooning the ball over the bar after being presented with an open goal.