It was the latest manoeuvre by the civilian opposition since Mali's military rulers failed to meet a March deadline to hold elections and hand over power to a civilian government.
"The citizen assembly of the civil transition has today elected the members of the government," read a statement datelined Geneva and signed by exiled Malian politician Adaman Traore, identified as the body's president.
This "civil transition (government)... is the only legitimate one in Mali," the text said.
It named the prime minister and defence minister of the rival government as Mohamed Cherif Kone, one of several prominent exiled politicians listed as members.
The announcement came a day after the political movement behind Mali's junta-appointed civilian prime minister, Choguel Kokalla Maiga, openly criticized the military rulers for the first time.
AFP was not able to confirm whether Maiga endorsed that position or Saturday's statement by the rival "government."
The colonels have kept a tight hold on power, suspending all party-political activities and muzzling opponents, journalists and human rights activists.
Mali has since 2012 been plunged into a political and security crisis fuelled by attacks from jihadist and other armed groups, as well as a separatist struggle in the north.