The 20-year-old Ivory Coast international joined City from Crystal Palace in 2010 but went on to make only seven appearances for the club, the last being as a late substitute in the 1-0 win over West Brom in the Premier League on 7 May.
Razak, who has spent spells on loan at Portsmouth, Brighton and Charlton during his time at City, joined Russian side Anzhi on a season-long temporary deal at the start of September.
A statement on the official City website read: "Due to the terms of the deal that took Abdul to Russia initially on loan, his transfer became permanent after his first appearance against CSKA at the end of September."
Meanwhile, City winger Alex Henshall has agreed to join Bristol Rovers on a one month emergency loan.
The 19-year-old, who has represented England at Under-16 and Under-17 level but has yet to make a senior appearance for City, was on loan at Chesterfield last season, where he made seven appearances.