Djibril Sidibe got the opener and Falcao found the net in fortunate fashion to double the principality club's lead inside five minutes at the Matmut Atlantique.
Falcao added two more goals in the second half, including one from the penalty spot, while Bordeaux ended a wretched evening a man down after the sending-off of Adam Ounas.
Europe's top scorers, Monaco have now scored 53 goals in just 17 league games, only four fewer than they managed in the whole of last season.
As a result they are now top of the table, above Cote d'Azur rivals Nice on goal difference, with defending champions Paris Saint-Germain four points back in third position.
PSG, who have run away with the title in recent seasons, are at home to Nice in a huge game on Sunday evening at the Parc des Princes.
"It was a great evening," said Monaco coach Leonardo Jardim.
"This year I think the league is more evenly balanced. Monaco are better, Nice too and I think Lyon are also going to come back. It's much more competitive and that is good for France.
"Our attackers have more quality and are making the difference. Our aim is just to keep progressing, that is the mentality."
Monaco have often struggled in Bordeaux down the years but they stunned their hosts by opening the scoring inside 80 seconds as Bernardo Silva set up France full-back Sidibe to find the net with a low finish across goalkeeper Jerome Prior.
It was 2-0 soon after when Prior tried to palm out a low cross from Valere Germain but the ball ricocheted off Falcao and into the net.
Any hopes Bordeaux harboured of a comeback in the second half were nipped in the bud as Monaco made it 3-0 on the breakaway in the 50th minute.
Thomas Lemar's long ball found Germain and he squared for Falcao, who took a touch before stabbing home.
The Colombian was handed the chance to complete his hat-trick from the penalty spot just after the hour mark when Nicolas Pallois petulantly threw Silva to the ground inside the box.
Falcao stepped up to score, taking his tally for the season to 10 goals in 11 league games and 14 goals in all competitions in what is a remarkable renaissance.
Jardim added: "He is a great goal-scorer. In the last two years things did not work out for him but this year, with the work he has put in, he has come back and is scoring goals."
Bordeaux's misery was completed when Algerian prospect Ounas was sent off for kicking out at Sidibe after fouling the Monaco man.
Earlier, Bafetimbi Gomis scored his eighth goal of the season late in the game as Marseille won 2-1 away at Dijon to register a first away victory of the campaign.
Maxime Lopez, 19, scored his first professional goal to put OM in front early on but ex-Newcastle United midfielder Mehdi Abeid had equalised for Dijon towards the end from a free-kick.
That match had been postponed on Friday evening due to thick fog in Burgundy and fog forced the postponement of the encounter between Nantes and Caen scheduled for later on Saturday.
Swedish international Ola Toivonen scored a hat-trick as Toulouse beat bottom club Lorient 3-2 in the mist, while Lille's revival under interim coach Patrick Collot continued as they beat Montpellier 2-1 for a third successive win.
Bastia climbed out of the relegation zone by beating Metz 2-0 and Nancy beat Angers by the same scoreline.