The unnamed man said he had slept for only two or three hours before the shift and confused the brake and the accelerator, leading to an "unexpected" loss of control, according to the video on the website of the local interior ministry.
"I do not know how it happened," he said, adding that he had not been drinking and was "very sorry".
The man, who authorities earlier identified as a 28-year-old national of the central Asian republic of Kyrgyzstan, said he ran from the scene fearing he would be killed there, before being taken into custody by police.
CCTV footage distributed on social media Saturday showed the yellow cab turning right as it tried to pass some standing traffic on a one-way street near Red Square.
The incident occurred about 200 metres (yards) from Red Square with Moscow packed with foreign football fans snapping pictures of the Kremlin.
Police said seven people had been injured but Russian news agencies quoted the city's health department as saying eight people were hurt.
Police closed the street for about an hour before reopening it to traffic.
Security has been a top concern heading into the June 14 to July 15 tournament.
At least 600,000 foreign tourists are expected to visit Russia during the most watched event in the world.
Source: AFP