Supporters of Julius Malema's Economic Freedom Fighters (EFF).
The protesters "put (a burning) tyre over the statue" of a soldier in the centre of the southern town of Uitenhage, police warrant officer Basil Seekoei told AFP.
"We haven't arrested anyone yet. We're still busy with the investigation first," he added.
Responsibility for the incident on Thursday was swiftly claimed by the EFF, a political party formed in 2013 by Julius Malema, formerly the firebrand youth leader of the ruling African National Congress, which expelled him after a conviction for hate speech.
Luxolo Jacobs, a self-proclaimed youth in the EFF, posted two photos on Twitter of the statue in flames and covered in plastic by party militants, with a message to Malema.
Police said that the memorial was "not badly damaged", since the stonework was merely blackened.
But the incident follows calls by Malema to bring down statues of South Africa's former white rulers, British and Afrikaaner alike.
"We said that economic liberation must be accompanied by the falling of these colonial statues and we would want to see them replaced by liberation hero statues," EFF Regional Deputy Chairperson Bo Madwara said on state-run SABC television.
South Africans are currently debating the status of colonial-era monuments more widely, after student activists at the University of Cape Town succeeded in having a statue of Cecil Rhodes boarded up.
Rhodes (1853-1902), the British colonist, mining magnate and politician for whom Rhodesia (modern Zimbabwe) was named, is seen in hostile circles as the embodiment of white oppression in southern African history.
The ANC, in power for 20 years following the end of white minority rule, has issued threats to colonial monuments but thus far left most of them standing in the name of national reconciliation. The party says it is open to discussion, within a legal framework.