At least five emergency vehicles were at the scene, Moscow police said.
CBS News are reporting that it is believed to have been an accident, as the driver may have been drunk at the time of the incident, and lost control of the vehicle.
One injured woman was in serious but not critical condition, according to Interfax, citing a medical source.
The Mexican Embassy in Moscow said on Twitter that two Mexican citizens were among the injured.
A witness said the driver got out of the auto, tried to run away but was caught by police, according to Reuters news agency.
CCTV footage posted on social media showed the cab driving about ten meters (yards) along the crowded pavement before hitting a traffic sign and coming to a partial stop.
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The driver had a license issued by Kyrgyzstan.
The incident occurred on Ilyinka Street, a short distance from Red Square, a popular tourist attraction in the Russian capital.
In central Moscow, authorities have installed heavy concrete blocks across the entrances to pedestrianised areas following a spate of incidents in European cities in which vehicles were used to mow down people.
Moscow Police have launched criminal investigation into a suspected violation of the traffic code.
At least 600,000 foreign tourists are expected to visit Russian Federation during the most watched event in the world. A total of 64 matches will be played in 12 venues in 11 Russian cities.