Claudio Ranieri’s side appeared to be heading to a draw in its first match back in the top flight, but substitute Emmanuel Riviere took advantage of a mistake by Bordeaux defender Marc Planus to score eight minutes from time and Falcao sealed the result five minutes later.
Monaco is level on points with defending champion Paris Saint-Germain – which drew 1-1 at Montpellier on Friday – after starting the season with a two-point deduction.
The French federation is currently deliberating Monaco’s appeal against the penalty, which saw the club deducted points after trouble flared at its final home game last season.
Despite being one of the three promoted sides, Monaco is widely expected to be the closest challenger to PSG this season. However, its return to the top flight almost started poorly as Bordeaux went close to taking an early lead, but new signing Ricardo Carvalho did enough to put off Henri Saivet, who fired over from close range.
Monaco started to get into its stride and could have gone in front in the 22nd minute but Ocampos’ powerful effort struck the outside of the right post.
The visitors had another good chance six minutes later as James Rodriguez did well to force his way into the area and carve out space for a shot but Bordeaux goalkeeper Cedric Carrasso pulled off a reflex save to push the ball over the crossbar.
Bordeaux improved in the second half and had several chances to take the lead. Saivet shot wide of the right upright from a difficult angle early on, and Jaroslav Plasil’s shot just whistled past the left post.
The home side was left to rue those missed opportunities as Monaco took the lead in the 82nd minute. Riviere – who had only come on 10 minutes earlier – made the most of a mistake by Planus to bring down an aerial ball and prod it into the bottom right corner.
Another defensive mistake shortly afterward allowed Falcao to beat Bordeaux’s Ludovic Sane to Lucas Ocampos’ cross and seal the match. The goal prompted wild celebrations from Falcao, who was signed from Atletico Madrid in an offseason deal worth an estimated 60 million euros.