The Colombian had missed twice from the spot this season but calmly converted on 43 minutes at the Stade de la Licorne after Emil Krafth upended Benoit Badiashile. Amiens defender Bakaye Dibassy was sent off for catching the Monaco captain with a raised boot in the 90th minute, and Falcao wrapped up a second victory in three games with another penalty deep into stoppage time.
Henry said he wasn't getting too carried away with his team's victory.
"From our side of things in terms of how we played it wasn't extraordinary but we held on. Fortunately Falcao was there when we needed him," Henry said.
"We must not overreact, we need to continue to work and use the positive aspects of each match to inspire us," he added.
The result provisionally bumped Henry's side up to 17th place above Dijon and Amiens on goal difference, confirming their recent upturn under the former France star in recent weeks.
Second-place Montpellier host Lille, in fourth, later on Tuesday as both sides attempt to cut into Paris Saint-Germain's 14-point lead at the top of the table. PSG will be without Neymar as they look to continue their undefeated record in Ligue 1 this season on Wednesday's trip to Strasbourg.
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