Mbappe's impressive double and Pogba's strike helped the 1998 world champions bounce back from Friday's 3-2 home defeat by Colombia, while Fedor Smolov's goal for Russia proved to be only a consolation.
The game was preceded by a minute's silence to remember the 64 people, including 41 children, who were killed in a fire at a Siberian shopping centre on Sunday.
Russia created the first clear chance of the match in the 15th minute when Alan Dzagoev found Smolov in the area, but Hugo Lloris kept out the Russian skipper's low shot from 12 yards out.
The visitors came close through Mbappe 10 minutes later, but the Paris Saint-Germain forward was denied by home goalkeeper Andrei Lunev.
But Mbappe put France into a 40th-minute lead when he latched onto Pogba's through ball, before cutting inside and lashing a low shot home despite the attentions of two Russian defenders.
Pogba doubled the advantage four minutes after the interval with a magnificent free-kick which curled into the bottom corner past the diving Lunev.
Smolov gave Russia hope, beating Lloris after meeting a cross from Igor Smolnikov.
But Mbappe restored France's two-goal cushion with seven minutes to go, skipping past Roman Neustadter and firing in a low shot which slipped through the grasp, and the legs, of Zenit St Petersburg keeper Lunev.
At the age of 19, Mbappe became the youngest player to score a brace for the France national team.
Although Russia put in an improved display from Friday's 3-0 loss to Brazil, Stanislav Cherchesov's side are now winless in five matches.