The police action against Mido Macia, 27, was "horrific, disturbing and unacceptable," President Jacob Zuma said.
"No human being should be treated in that manner," he said.
The officers, who were accused of being involved in the incident, were arrested Friday, CNN reported.
All were disarmed and suspended from their duties.
"We view this incident in serious light. We want stern action so that it may send a message to other officers that any untoward conduct will not be tolerated," State Security Minister Siyabonga Cwele said.
It was unclear how many officers were involved in the Tuesday night incident in Daveyton, a township about 27 miles southeast of Johannesburg.
Cellphone footage posted online shows the man, later identified as Macia, a Mozambican taxi driver, talking with police officers, then being handcuffed to the van, which pulls away, dragging his feet along the road. Officers and bystanders run alongside as the van increases its speed.
Macia later died in a police cell, officials said.
"We are shocked by this incident," said Moses Dlamini, a spokesman for the Independent Police Investigative Directorate, an independent government agency investigating the incident.
The IPID, created in 1997 as part of post-apartheid reform of South African police, said its initial investigation indicated police asked Macia to move his taxi because it was obstructing traffic at a marketplace.
"This appalling incident involving excessive force is the latest in an increasingly disturbing pattern of brutal police conduct in South Africa," said Noel Kututwa, Amnesty International's southern Africa director.