The Colombian is now the league's leading scorer ahead of team mate Emmanuel Riviere who opened the scoring in the 39th minute with his sixth of the season.
Falcao doubled the lead two minutes later when Joao Moutinho, another multi-million signing, set him up with a deep pass over the defence.
He wrapped it up from close range one minute from time as Monaco moved on 17 points from seven games.
They are two points ahead of champions Paris St Germain who won 1-0 at Valenciennes thanks to a rare combination between Zlatan Ibrahimovic and Edinson Cavani.
Uruguayan Cavani scored in first-half stoppage time with a clinical finish after his Swedish team mate set him up following a 50-metre run with the ball as PSG made the most of their one clear-cut chance in a dull game.
It was the first pass between the two players, who were described as Europe's best attacking pairing by former sports director Leonardo when Cavani joined this year but who have struggled to gel so far.
Last season's runners-up Olympique Marseille are third on 14 points thanks to a 2-1 victory over St Etienne on Tuesday.