A Reuters photographer at pitch level for France’s 3-1 win over Russia heard monkey chants directed at French players on several occasions, including when midfielder N’Golo Kante came to the touchline for a throw-in.
FIFA said it had conducted a "thorough investigation" of video evidence of the match in reaching its decision.
World soccer's ruling body said it recognised the gravity of the incident but added that only a limited number of fans were involved.
The Russian Football Union (RFU) referred questions to its anti-discrimination inspector Alexei Smertin, whose assistant declined to comment when contacted by Reuters.
Russia has pledged to crack down on racism as the country faces increased scrutiny ahead of this year’s World Cup, which it will host from June 14 to July 15 in 11 cities including Moscow, St Petersburg and Sochi.
There have been other reported cases of racist abuse, however, in the run-up to the tournament.
Alexei Sorokin, chief executive of Russia's local World Cup organising committee (LOC), was reported by TASS news agency as saying that such incidents were rare.
"It is a shame that such discriminatory incidents take place here, but thankfully they are very rare and do not constitute a trend," he said.
"We have gone quite a long way to isolate this problem. It is impossible to control such human reactions in any country in the world."
FIFA confirmed in its statement that it would use its three-step procedure at the World Cup in the case of racist incidents.
This gives referees the authority to stop matches and request a public announcement warning that all discriminatory behaviour must stop.
If the trouble continues, the referee can halt the match and ask for another warning and, if it still carries on, the game can be abandoned.
(Reporting by Gabrielle Tetrault-Farber; writing by Brian Homewood, editing by Ed Osmond)
