The Colombia international headed home right on on half-time, after Serge Gakpe had earlier seen a spot-kick saved by Danijel Subasic
The Colombia international, back in the starting line-up after his recent return to full fitness following a knee injury, headed home just before the break and Leonardo Jardim's men held on by the skin of their teeth to leave their hosts counting the cost of Serge Gakpe's 23rd-minute penalty miss.
Monaco arrived in Nantes still reeling from the shock of losing their first two games of the new Ligue 1 season but they did not look short on confidence in the early exchanges, dominating possession and forcing a succession of corners in the opening quarter.
By complete contrast, Nantes struggled to string more than a couple of passes together and Yacine Bammou cut an isolated and frustrated figure up front.
However, the striker made his presence felt 23 minutes in when he escaped the attention of Andrea Raggi before being felled in the area by Subasic.
The Monaco goalkeeper was only booked for his indiscretion and that proved crucial, because he brilliantly saved Gakpe's subsequent spot-kick before Monaco eventually managed to scramble the ball to safety after a game of pinball in the six-yard box.
It proved a costly miss for Nantes, who fell behind just before the break when the lively Layvin Kurzawa whipped over a wonderful left-wing cross that Falcao headed home emphatically at the back post - despite the best efforts of Maxime Dupe, who got a hand on the ball.
Monaco could have doubled their lead in the early stages of the second half but Dupe proved equal to Lucas Ocampos' well-struck shot from the edge of the area.
Nantes dominated thereafter, though, initially failing to create any clear-cut openings before piling on the pressure in the closing stages, with Bammou dragging a half-volley just wide of Subasic's goal before Kian Hansen crashed a header against the crossbar with just five minutes remaining.
The hosts kept coming in injury time but Subasic, rather fittingly, had the final say, palming Itay Shechter's downward header up and over the crossbar to earn his side - and his coach - three much-needed points.