His new side beat Montpellier 4-1 to maintain a 100 per cent record in Ligue 1, as rivals Paris Saint-Germain slipped up.
After opening his account for the principality in last week's victory at Bordeaux, the Colombian kicked off a comfortable win over 2012 champions Montpellier with a penalty at the Stade Louis II.
Victor Montano soon equalised but Emmanuel Riviere hit a hat-trick to guarantee three points and keep Monaco tied for top spot with Lyon, Marseille and St Etienne.
However, reports in Spain last week suggested 60million euro striker Falcao was unhappy with the scope of the project on the south coast after leaving Atletico Madrid in May, with a controversial switch to his former club's fierce rivals Real Madrid mooted as a surprise option.
The 27-year-old rejected the reports out of hand, though, telling L'Equipe: "It's a totally false article. I'm very happy at Monaco and I believe in the project.
"Everything is false, even the numbers quoted by the press are false. I feel great at the club and I'm going to stay here."
Monaco boss Claudio Ranieri also gave the reports similar short shrift, adding: "What do you want me to tell you? The rumours are rumours.
"Falcao is staying at Monaco."