No details of the length of contract or transfer fee were revealed.
Zenit have already signed defender Igor Smolnikov from Krasnodar as they looked to strengthen their squad ahead of the Champions League play-off tie against Portugal's FC Pacos de Ferreira on Tuesday.
"Everything turned out unexpectedly but this is the football life," Shatov told the official UEFA website.
"I got a nice offer. Of course, I want to represent a team with ambitious aims."
After losing their opening four games of the season, Anzhi announced that their entire squad was for sale, signalling a change in strategy from signing big-name players.
Chairman Konstantin Remchukov also took to Twitter to suggest the club's annual budget will be reduced from $187million (£119m) to around $78 (£50m) million.
Since the announcement the futures of players such as Brazilian playmaker Willian and Cameroon striker Samuel Eto'o have been in doubt with Willian linked to a move to Premier League Tottenham Hotspur and Eto'o to Chelsea.