The arrival of Mutko, who also oversees Russia's World Cup preparations, comes as Swiss prosecutors investigate alleged corruption in the bidding process for the 2018 tournament.
Mutko has been a member of FIFA's executive committee since 2009 and played a leading role in Russia's winning bid.
Mutko was previously Russian Football Union president from 2005 until 2009 and takes over from Nikolai Tolstykh, who was ousted in May after disputes over the organization's debts.
"We need to organize a precise system of running football, in order to discover the potential of our football, which is there. That's the most important strategic goal," he said in comments reported by the R-Sport agency.
Mutko also talked up the possibility of shoring up the RFU's finances with a possible 3 billion rubles ($44 million) in new sponsorship money, with a first deal to be signed as soon as Thursday morning, R-Sport reported.