"If we do not do it, it would be a catastrophe," Mutko told Russian state television on Saturday.
The host of the 2018 FIFA World Cup, Russia lost its last European qualifier 1-0 at home to Austria and is currently third in Group G.
The match against Sweden on 6 September is shaping as a must-win duel if it is to retain any chance of automatic qualification.i
Mutko again said that sacking Capello was an option but that step would require Russia to pay out the 69-year-old Italian's contract, valid to 2018, in full.
Unconfirmed reports have said Capello earns nearly $US8 million a year.
Many fans and critics in Russia complain Capello has not been able to adapt his tactics to modern football while his backers retort he does not have the players available to allow him to do so.