The attacker, on loan from AC Milan, saw his penalty saved but hit home the parried effort to cancel out Layvin Kurzawa's strike earlier in the second halfMonaco were held to a 1-1 draw by Montpellier on Friday after M'Baye Niang returned to Ligue 1 in style by scoring the crucial equaliser.
Layvin Kurzawa had put the visitors ahead with an unorthodox shouldering of the ball from James Rodriguez's free kick, which looped into the top corner to put Laurent Nicollin's behind at the Stade de la Mosson.
Niang, on loan from AC Milan, made a goalscoring debut for la Paillade in midweek in the Coupe de France and made it two goals in as many games by converting the rebound to his initial penalty to deny Monaco their fourth Ligue 1 away win in a row.
The hosts were dominant for much of the first half but were fortunate to go in level at the break after Remy Cabella fluffed his lines one-on-one with les Rouge et Blanc goalkeeper Danijel Subasic.
The second half was a more balanced affair, with goals at either end and half-chances galore from both sets of players, and a draw was a fair result in the first Ligue 1 game of 2014.
Pre-match favourites Monaco unquestionably started the stronger of the two sides. Moutinho was unlucky not to get his name on the scoresheet with a curled shot from 25 yards, as goalkeeper Geoffrey Jourdren reacted at the last millisecond to beat the shot behind.
The game fell into a lull as Monaco pressed the hosts but failed to finish off their finely-crafted chances. Falcao, in his first Ligue 1 start since November, looked rusty with his finishing as his efforts either sailed over, wide or dribbled harmlessly into the goalkeeper's arms.
Montpellier were mostly limited to passing around in the middle of the park in the first third of the match, though Cabella was a bright spark as he unsuccessfully called for a penalty in the 34th minute after claiming he was dragged down in the box by Kurzawa.
Cabella then should have put the hosts ahead with half-time lingering, after Eric Abidal played him onside, but the French attacker got the ball tangled in his feet and fumbled the shot, allowing Monaco shot-stopper Danijel Subasic to pounce.
After the break, Ranieri's men remained under pressure as la Paillade looked to prevent the visitors from leaving with a point or three. A majestic diving interception from Jeremy Toulalan denied Montpellier captain Hilton from essentially tapping in five minutes into the second half.
Against the run of play, Monaco sneaked into the lead on 52 minutes. Rodriguez's gorgeous free kick was looped into the top left-hand corner from 10 yards off Kurzawa's shoulder.
The hosts reacted strongly and Cabella finally got the penalty he had craved all game after tricking Monaco skipper Abidal into tripping him up 10 yards from goal.
Subasic initially saved Niang's shot but the Milan loanee followed up to convert the spot kick and level the score with 22 minutes to play.
Les Rouge et Blanc almost grabbed their second goal at the Stade de la Mosson within 60 seconds of the restart, but Emmanuel Riviere's diagonal through ball after bursting into the box trickled inches too far ahead of Falcao's outstretched leg.
At the other end, with 15 minutes to go, Cabella hit a pot shot straight at Subasic, as the home fans found full voice in anticipation of a potential upset.
However, it was not to be and both teams shared the spoils - with Montpellier rooted still to fourth-from-bottom while Monaco sit two points adrift of leaders Paris Saint-Germain having both played a game more than every other team in the division.