Falcao was absent after being hospitalised with a blow to the head in the principality club's 4-0 trouncing at local rivals Nice during the week.
And Leonardo Jardim's side struggled to get the better of in-form Angers, a Dickson Nwakaeme own goal proving the difference after Kamil Glik had cancelled out Famara Diedhiou's opener.
The result allowed Monaco to go two points clear of Nice, who visit Nancy on Sunday, and Toulouse, conquerors of champions Paris Saint-Germain on Friday.
Falcao out of hospital after 'rugby tackle'
Youngsters Abdou Diallo and Kevin N'Doram started for Monaco with Jardim having one eye on Tuesday's Champions League encounter with Bayer Leverkusen.
Angers were chasing a fourth consecutive win and they thought they had taken the lead early in the second half when Karl Toko-Ekambi had the ball in the net but the offside flag cut short the celebrations.
However, the visitors did go in front in controversial circumstances in the 56th minute as leggy Cameroon forward Toko-Ekambi slid the ball across goal for Cheikh Ndoye.
The Angers captain made a mess of his shot and the ball came up and struck his hand before falling for Diedhiou, who scored. The goal stood despite the appeals for a handball.
Balotelli brace helps send Nice top of Ligue 1
Nevertheless, Poland defender Glik fired in a 66th-minute equaliser from eight yards after Angers struggled to clear and the hosts secured the points with quarter of an hour left.
Mathieu Michel saved from Glik but the ball rebounded off Nigerian substitute Nwakaeme and in.
Monaco's eighth win in nine games in all competitions came after Lyon missed the chance to move level on points with stumbling PSG as they lost 1-0 at struggling Lorient.
Jimmy Cabot scored the only goal of the game early in the second half in Brittany as Lyon lost for the third time in the league already this season.
Bruno Genesio's side, who visit Sevilla in the Champions League this coming midweek, had beaten Montpellier 5-1 on Wednesday but that is now their only win in five Ligue 1 games.
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"I am not worried," said Genesio, whose side were again without injured leading scorer Alexandre Lacazette and suspended captain Maxime Gonalons, although France midfielder Mathieu Valbuena came in for his first start since April and hit the bar in the first half.
"There is no need to panic after this result.
"I often say that the result is one thing but there is also the performance, whether we win or lose.
"We need to keep calm. It is a shame, because it was a chance to climb the table and I think we deserved better."
Champions PSG slumped to a 2-0 defeat at Toulouse on Friday with Yann Bodiger scoring a penalty and Jimmy Durmaz also netting for the hosts after Serge Aurier had been sent off for the team from the capital.
PSG are fourth, three points behind Monaco and level on points with Bordeaux and Metz.
Bordeaux were held to a 0-0 draw by Caen while newly-promoted Metz continued their good start to the campaign by winning 1-0 in Montpellier thanks to a Mevlut Erding penalty.
Bastia defeated Guingamp 1-0 and Lois Diony scored twice as Dijon beat Rennes 3-0.
Lille are bottom of the table before they go to Saint-Etienne on Sunday, with Marseille also in action at home to Nantes.
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