Video of the incident posted on social media shows the vehicle veering out of standstill traffic and accelerating onto the sidewalk, mowing down pedestrians.
A taxi cab drove onto the pavement and ploughed into a crowd of pedestrians on central Moscow's Ilyinka street.
Bystanders quickly go to the driver's side door and try to force it open.
Ambulance and police work at the site of an incident after a taxi crashed into pedestrians on a sidewalk near Red Square. The authority cited the driver as saying he had not driven into the crowd on objective.
Eight people have been injured after a taxi rammed through a crowd in Moscow.
Moscow police said the driver was a 28-year-old national of Kyrgyzstan.
Police later released a video of the suspect being questioned about the crash.
An eyewitness account from the police station published on a Russian Telegram channel showed the driver sitting handcuffed on a bench and looking dazed.
"The sense we are getting is that (the taxi crash) is not football-related", Mark Roberts, head of United Kingdom soccer policing, told the AP.
The accident took place on Ilinka Street, around 200 metres from Red Square and Moscow's famous GUM shopping centre.
Russia's state-run TASS news agency, citing a source with emergency services, said that two Mexicans, two Russians, and one Ukrainian citizen were hospitalized following the incident.
The damaged taxi is towed away from the scene. Among security concerns has been the use of vehicles as weapons.
Saturday's incident has focused attention on security at the planet's most watched event. The fans were walking below him on their way back from Spartak Stadium, where Argentina tied Iceland 1-1 in their first match of the competition.