The road was closed to facilitate access to the shrine, which is a popular destination for people on New Year's Eve.
Police say the man is suspected of intentionally trying to kill people by ploughing the medium-sized van into crowds who had lined the streets to welcome in 2019.
Tokyo Metropolitan Police Department has yet to issue a statement on the matter.
The vehicle barreled down the narrow street, which was packed with pedestrians, slamming into a number of them, police said.
"I learned about the incident later".
A BBC producer who was a witness at the station described the attacker as wielding a large kitchen knife and who linked his actions to "bombings in other countries". "It was very very scary", Clack said.
Bomb attack kills 3 tourists, one Egyptian
The interior ministry confirmed the death of two of the tourists, and the state prosecutor's office later said a third had died. But that figure was still far short of the record influx in 2010 when over 14 million visitors flocked see the country's sites.
At least nine people were hurt shortly after midnight on Tuesday when a man rammed his vehicle into pedestrians on a Tokyo street in what he claimed was an act of terrorism, local media reported.
NHK said the suspect had initially described the incident as an "act of terror" but later said the attack was in retaliation for capital punishment. The spokesman declined to elaborate.
Firefighters carry on rescue works after a auto crash in Tokyo, Japan, Jan. 1, 2019.
The crash occurred early on Tuesday (local time) on Takeshita Street, well known to tourists and pop culture and fashion fans.
"I can't believe it, this is a place I'm familiar with, so it's very shocking", said Tatsuhiro Yaegashi, a 27-year-old worker in the area.