The striker has penned a four-and-a-half-year contract with Les Rouge et Blanc but there is speculation that he could be immediately loaned to West Ham for the rest of the season
The Cote d'Ivoire international only joined the Russia outfit in June 2012 for a reported €18 million (£14.9m), but has become the latest high-profile departure from Dagestan after club owner Suleyman Kerimov announced severe cost-cutting measures last summer.
"Lacina Traore, 23, has signed a four-and-a-half-year contract with AS Monaco FC," read a statement on the Ligue 1 side's official website.
However, it has not yet been confirmed whether the striker will, as rumoured, be immediately loaned to Premier League strugglers West Ham United for the remainder of the 2013-14 season.
Monaco - who only achieved promotion back to the French top flight last season - are currently second in the standings, just three points behind leaders and reigning champions Paris Saint-Germain.