Russia is now hosting the World Cup and Moscow is one of 11 Russian cities where tournament games are being played. Authorities say the driver lost control of the vehicle.
"According to preliminary estimates, seven people were injured", the source in the city's emergencies services told TASS.
Argentina took on Iceland at the Sparatak Stadium in Moscow earlier today.
The driver of the taxi has been detained, officials said.
The driver has been arrested after reportedly trying to flee the scene.
Witnesses say the man was shouting "it wasn't me" when he was being pulled out of the vehicle before running away.
The driver said the incident was an accident, traffic authorities said in a subsequent tweet.
Moscow's traffic monitoring agency said the driver claimed the crash wasn't premeditated.
Video of the incident posted on social media shows the vehicle veering out of standstill traffic and accelerating onto the sidewalk, mowing down pedestrians. There are no fatalities but at least one woman has serious injuries although she is not believed to be critical, according to Interfax.
Russian authorities have promised to host a safe World Cup tournament, which is taking place in 11 cities until 15 July.
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