The Tass news agency said that two Mexicans, two Russians and one Ukrainian were hospitalised after the crash.
"The driver failed to control the vehicle", Moscow Mayor Sergei Sobyanin said.
Police have arrested the driver, who is from Kirguistan, despite his attempts to flee.
One injured woman was in serious but not critical condition, according to Interfax, citing a medical source.
The video then shows witnesses run over to the driver's side door and yank the young male driver out of the vehicle.
The yellow cab had mounted the pavement in Moscow and ploughed straight into a group of pedestrians.
Interfax news agency cited a source as saying that the driver was drunk at the time of the incident.More news: Portugal vs. Spain: 3 things we learned
More news: The DOJ inspector general's report: Hillary Clinton has good reason to complain
More news: Here are the details on the Wilkes-Barre tornado from National Weather Service
Mexican nationals were among the injured, it said.
City traffic authorities posted a photograph of what they said was the driver's Kyrgyz driver's license showing that he is 28-year-old native of the Central Asian nation.
Authorities say taxi driver is in custody after crashing into crowd in Moscow as Russian Federation hosts the World Cup.
A witness said some of the people struck by the taxi were wearing Mexican team colors, Reuters reported.
The crash occurred near Gostiny Dvor, where visitors have gathered amid World Cup festivities.
The US State Department on Friday updated its travel advice on Russian Federation, saying terrorist groups were plotting attacks.