Mankiev, 23, who was competing in the 55kg category, beat the former-European champion Rovshan Bairamov from Azerbaijan in the final to take the gold. Armenia's Roman Amoyan and South Korea's Park Eun-chul took the bronze medals.
In the 60kg event Albiev also faced an opponent from Azerbaijan, Vitaly Rakhimov, but it was the Russian who proved the stronger to win Russia's second gold of the games.
Kyrgyzstan's Ruslan Tumenbaev and Kazakhstan's Tengizbayev Nurbakyt took the bronze medals.
The wrestling victories have propelled the Russian Olympic squad into 7th place in the overall medal standings with 9 medals - 2 gold, 4 silver and 3 bronze. Russia will be hoping for more medals as the wrestling competition, one of the country's stronger events, develops in the coming days.
The standings are topped by the Olympic hosts, China, which boasts 18 medals - 11 gold, three silver and four bronze.