Manchester United's Paul Pogba and Barcelona's Ousmane Dembele are said to be among those subject to insults.
The match was played the Krestovsky Stadium in Saint Petersburg, a host scene at this present summer's World Cup.
"Bigotry has no place on the football pitch," said French Minister for Sport Laura Flessel.
"We need to act as one at an European and universal level to put a conclusion to these forbidden activities."
The impressive Pogba then won and lined up a free-kick on 49 minutes, curling over the wall and into the bottom corner past Lunev.
Sub Igor Smolnikov's fine ball across the face of goal was then turned home by Smolov at the far post to create some anxiety in the French defence in the closing stages.
But Mbappe, who showed superb skill to find a yard in the box, saw his weak shot roll through the legs of Lunev on 83 minutes to make sure of the win