The Colombia international netted with just three minutes to go in his first game for the victorious Rouges et Blanc, while Lyon also made a winning start to their season
The Colombian striker had to wait until the final three minutes to get his first strike for the club, reacting quickest to control a loose ball before squeezing his shot under Cedric Carrasso from close range.
Falcao's strike came five minutes after substitute Emmanuel Riviere had broken the deadlock with a left-footed volley shortly after replacing James Rodriguez.
Lyon cruised to a 4-0 victory over Nice at Stade Gerland, with Alexandre Lacazette scoring twice for the seven-time French champions. Clement Grenier and Yoann Gourcuff also got their names on the scoresheet as Remi Garde's side made short work of the team they finished one place and three points ahead of last season.
Like Monaco, fellow Ligue 1 newboys Nantes started their campaign with a victory. Goals from Filip Dordevic and a Julian Palmieri own goal gave Michel Der Zakarian's side a 2-0 win against Bastia, with both sides having a man sent off after the break.
Toulouse had a disappointing start the season with a 3-0 defeat at Valenciennes. Uros Spajic's sending off just after the half hour gave the hosts a penalty that Maor Melikson converted, with Jose Saez and Gregory Pujol sealing the three points in the second half.
There were also opening-day victories for Lille and Rennes over 10-man Lorient and Reims, respectively, while Evian were held to a 1-1 draw by Sochaux after Roy Contout's strike cancelled out Fabrice Ehret's opener.
On Friday, Ligue 1 champions Paris Saint-Germain could only secure a 1-1 draw at Montpellier.