Cenk Tosun scored twice for the visitors, netting either side of the interval, after Radamel Falcao had opened the scoring for the Ligue 1 club.
The result put Turkish side Besiktas on a perfect nine points from three games and left Monaco propping up the table with a solitary point.
RB Leipzig are second on four points after a 3-2 win over Porto, who are a point further back in third.
Monaco, who have now failed to keep a clean sheet in their previous 11 Champions League games went ahead on 30 minutes when Colombia striker Falcao cleverly slid the ball past Besiktas keeper Fabri from inside the area.
Besiktas, however, reacted swiftly and levelled four minutes later when Tosun headed home from Ricardo Quaresma's cross.
Nine minutes into the second half, Ryan Babel perfectly set up Tosun in the box and the Turkey striker clinically beat Monaco keeper Danijel Subasic.
Besiktas then held on to claim their first win on French soil in five matches.
(Reporting by Julien Pretot; Editing by Toby Davis)
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