The Colombia international made his long-awaited return from the knee ligament problem that ruled him out of the World Cup with a substitute appearance against Valencia on Saturday, and headed in the opener just after the half-hour mark.
With Valencia beating Benfica 3-1 earlier in the day at the Emirates Stadium, Arsenal's defeat denied them the chance to regain the pre-season tournament title but Arsene Wenger will have been disappointed with Martin Atkinson's baffling decision to award a free-kick instead of a penalty when Chuba Akpom was fouled in the area in the closing stages.
On a positive note, Wenger was able to hand a debut to Mathieu Debuchy while seemingly avoiding any injuries ahead of next week's Community Shield.
Wenger was given a scare early on when Olivier Giroud went down clutching his back, but the striker was able to continue in what was a sedate opening to the match.
Giroud and a supporting cast of Alexis, Aaron Ramsey and Santi Cazorla failed to inspire a sluggish Arsenal while Monaco's closest attempt early on was a wayward Yannick Ferreira Carrasco cross that almost caught out Wojciech Szczesny.
Joao Moutinho almost opened the scoring when he nudged Lucas Ocampos' delivery wide before Ferreira Carrasco then let fly with an effort that just crept over the crossbar.
After a tentative start, the chances then began to arrive as Jack Wilshere forced a save from Danijel Subasic before Falcao put Monaco ahead.
Wilshere conceded a free-kick following a late tackle on Nabil Dirar before Colombian forward Falcao lost his marker to nod home Moutinho's delivery, scoring his first goal since January.
Subasic was then required to keep Wilshere's strike out with a strong palm after nifty work from Alexis as Arsenal started the second half well.
Alexis - moved into the centre after Alex Oxlade-Chamberlain had replaced Giroud at half-time - looked increasingly threatening as he fired wide from inside the area following a lay-off from Cazorla.
With Falcao, Moutinho and Alexis all taken off midway through the second half, both sides lost a degree of quality in the attacking third as Monaco were able to see out the win.
Leonardo Jardim's side were given a let off when referee Martin Atkinson surprisingly awarded a free-kick rather than a penalty for Subasic's foul on Chuba Akpom, despite the contact appearing to have been in the box, but the French side will go into their Ligue 1 opener against Lorient next Sunday off the back of a win.
For Arsenal, a lack of cutting edge ultimately cost them - in contrast to Saturday's 5-1 win over Benfica - with Wenger sure to demand an improvement in the Community Shield curtain-raiser against Manchester City in seven days' time.