President Vladimir Putin signed a decree appointing Nebenzya, the Kremlin wrote on its website.
Nebenzya will start in the new post as relations between Russia and the West have hit post-Cold War lows over Moscow's annexation of Crimea from Ukraine and its role in escalating the eastern Ukrainian separatist conflict.
Churkin, who died suddenly after collapsing at work aged 64, was a well-known figure who had been Russia's UN ambassador since 2006. In recent years he energetically defended Russia's policies on Ukraine and Syria.
Churkin's successor, 55-year-old Nebenzya, is a career diplomat who previously served as a deputy foreign minister. He will also represent Russia at the UN Security Council.
Nebenzya began his career as an attache at the Soviet embassy in Thailand. He previously worked at the UN from 1996 to 2000 as a senior advisor to Russia's mission in New York. He speaks English and Spanish, according to the foreign ministry website.