The 19-year-old Mbappe became the youngest France player to score an away goal since Rene Gerard found the back of the net in Germany in 1933.
Coach Didier Deschamps admitted France were counting heavily on Pogba, saying: "We expect a lot from him, I ask a lot from him".
Federation Internationale de Football Association have launched an investigation after monkey chants were directed at Manchester United star Pogba, Barcelona's Ousmane Dembele and Chelsea's N'Golo Kante during France's 3-1 win over Russia in St Petersburg's Krestovsky Stadium on Tuesday.
With CSKA and Spartak defenders Viktor Vasin and Georgi Dzhikiya out with injuries as well as Zenit's forward Alexander Kokorin, Russia will likely once again depend on Fyodor Smolov up front.
The French could have scored more, particularly Anthony Martial, who failed to convert good chances in the first half, while defender Vladimir Granat missed an open goal for Russian Federation at a corner.
According to Duncan Castles for Yahoo Sports, Mourinho wants next term to be the best season of Pogba's career and believes the Frenchman must apply himself more having been poor in training of late as well as on the pitch.
"He was also playing a "comeback" match because it's been a while since he played a full game".
But France had the final say as Mbappe added his second, firing through the legs of Russian Federation stopper Lunev.
Russia, who were humbled 3-0 by Brazil on Friday, did little to show they could go past the group phase at the World Cup after failing to make the knockouts in their last three appearances at major championships.
France kick off their World Cup campaign against Australia on June 16.