The Colombia striker netted with a powerful first-half header to give Monaco victory in their final pre-season friendly before they open their Ligue 1 campaign at home to Lorient next Sunday.
Arsenal manager Arsene Wenger handed a full debut to £30 million ($50.4 million, 37.6 million euros) new signing Alexis Sanchez, but the Chilean was eclipsed by Falcao, who missed the World Cup after damaging knee ligaments in January.
Monaco’s win handed victory in the four-team tournament to Valencia, who beat Benfica 3-1 earlier in the day at Arsenal’s Emirates Stadium.
Arsenal tackle Manchester City in the Community Shield at Wembley next weekend, before kicking off the Premier League season at home to Crystal Palace on August 16.
Sanchez lined up in front of debutant Mathieu Debuchy on Arsenal’s right flank, with fellow new signing Calum Chambers again deployed at centre-back, while Falcao made his first start since his injury.
Buzzing around in bright orange boots, Sanchez almost freed Olivier Giroud with a sweeping pass from wide on the right, but while Arsenal looked bright and eager in possession, Monaco carried the greater threat.
Right-back Nabil Dirar almost embarrassed Arsenal goalkeeper Wojciech Szczesny with a mishit right-wing cross and then teed up Falcao to head wide.
Yannick Ferreira-Carrasco twice threatened from range, drilling narrowly over from 25 yards with his right foot and then working Szczesny with his left, while Joao Moutinho also shot wide.
– Falcao unmarked –
All that Arsenal could offer by way of reply was a Chambers effort from a Santi Cazorla corner that sailed well clear of the crossbar.
Monaco made the breakthrough in the 37th minute when Falcao stole in unmarked to meet a free-kick from Moutinho with an emphatic close-range header.
It was the former Atletico Madrid striker’s first goal since the day he ruptured the anterior cruciate ligament in his left knee in a French Cup tie against amateur side Chasselay on January 22.
A half-time reshuffle saw Sanchez move up front for the second half and he played a role in a move that culminated in Monaco goalkeeper Danijel Subasic swatting away a thumping shot from Jack Wilshere.
Falcao made way for Anthony Martial with around half an hour to play, while Sanchez was also withdrawn before the end.
Teenage striker Chuba Akpom took Sanchez’s place and he should have been awarded a penalty after being tripped by Subasic, but referee Martin Atkinson instead gave a free-kick on the edge of the box that Arsenal could not exploit.
In the penultimate game of the tournament earlier on Sunday, Valencia came from behind to beat Portuguese champions Benfica 3-1.
New signing Derley gave Benfica an early lead, but Jose Luis Gaya equalised just before half-time and Valencia completed their comeback with goals from Pablo Piatti and Andres Guardado in the second half.