Seven or eight people were injured during the crash, according to local media reports.
The driver, who is believed to have had a Kyrgyzstan driving licence, attempted to flee but has now been detained by officers.
"The driver failed to control the vehicle", Moscow Mayor Sergei Sobyanin said Saturday.
The city health department said seven of the injured were in satisfactory condition and one was in more serious condition.
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.
White House says Trump supports both House immigration bills, after all
There are two bills slated for consideration next week: the Republican compromise bill and the conservative Goodlatte-McCaul bill. The comments came as House Speaker Paul Ryan and other Republicans said they were not comfortable with family separations.
Moscow's traffic management authority said the driver was not drunk and Interfax news agency cited a source saying there was no alcohol in his blood.
The Moscow traffic authority has said the taxi driver said he was not acting on goal.
Witnesses said some of the victims were wearing the green football colours of Mexico ahead of the team's match against Germany in the city tomorrow.
Two Mexican nationals, who had come to Russian Federation for the World Cup, were among the injured in the incident.
At least 600,000 foreign tourists are expected to visit Russian Federation during the most watched event in the world.
Russian authorities have pledged to stage a safe World Cup.
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