The rescue operation, involving Russia's special FSB forces and police agents, ensued when the hijacker agreed to allow medics aboard to treat passengers who had fallen ill.
Doctors went aboard, accompanied by police dressed as medics, and the man was overpowered, said Tatyana Trunova, a spokeswoman for transport police.
"It was the fastest aircraft-freeing operation in the history of civil aviation," said Yelena Galanova, an airport spokeswoman.
Russian news agencies cited police officials as saying the man was not armed.
Mineralniye Vody is in Russia's troubled North Caucasus region, which suffers daily violence attributed to separatist fighters.