The Colombian forward was the subject of interest throughout last season from major clubs including Manchester United, Chelsea, Real Madrid and Paris St Germain, but in the end opted for the Ligue 1 outfit from the Mediterranean Principality, owned by Russian billionaire Dmitry Rybolovlev.
The move was made official on May 31, after Monaco paid Atletico Madrid an estimated 60million euros and agreed a five-year contract with the player.
Monaco, coached by Claudio Ranieri, were promoted to the French top flight last season and will not be competing in European competition in 2013-14.
And in his first press conference as a Monaco player on Tuesday, Falcao acknowledged that eyebrows had been raised at his move.
He said: "Many people think that coming here may be an error and perhaps they are right. But when we win a trophy, I'll know it was the right decision.
"Choosing a club that isn't in Europe may have surprised some people but we want to be in the Champions League next season. That's our motivation."
He explained he had moved to the Ligue 1 side "for a number of reasons".
"The president and his coaching staff have the same ambition as me, which is to reach the top of European football," he said.
"They assured me that I was as a fundamental part of this project, which gave me confidence and security.
"This club wants to win trophies and that, together with the role I'll have in the team, made me want to come here."
Monaco's stadium, the Stade Louis II, holds 18,500 spectators and last season had an average attendance of just 5,086.
Falcao hopes the improving team will be able to attract more fans to the stadium, but emphasised that there were more important things than playing in front of big crowds.
"We will have to motivate fans to come and support us in our stadium," he said.
"[But] you don't get motivated because of the fans in the stands, you get motivated by beating your rivals. And we are stimulated by the fact Monaco want to win the league and get into Europe. That's what should guide us.
"The points are worth the same with or without people in the stands."
Falcao will play a leading part in a squad that includes compatriot James Rodriguez, plus Portuguese players Joao Moutinho and Ricardo Carvalho. The club announced the signing of Eric Abidal yesterday while Jeremy Toulalan joined last weekend.
Toulalan and Abidal plied their trade in La Liga for the last two seasons and came up against Falcao, who believes his new club will have a finely balanced squad.
"I played against Toulalan and Abidal and they are magnificent," he said.
"They are experienced and can help the younger players. This team will not just be good in attack, it will also be solid in defence."