Zeid Raad Al Hussein described the incident, in which 10 people were wounded, as "extremely shocking".
Cizre has been the focus of an offensive by Turkey's army against the Kurdish rebel group, the PKK.
Mr Al Hussein also expressed concern over reports that the man who filmed the incident was arrested.
"Filming an atrocity is not a crime but shooting unarmed civilians most certainly is," he said.
"It is essential that there is a thorough, independent, impartial investigation into this and any other events that have led to the wounding or killing of civilians."
The incident in Cizre, a predominantly Kurdish city, occurred on 20 January. Video footage posted online purporting to be of the immediate aftermath of the shooting shows people being loaded into ambulances amid screaming and the sound of explosions.
Reports suggested that the group who came under fire were attempting to rescue people wounded in an earlier clashes.
The video footage also appears to show that some of the group were pushing a cart loaded with bodies across the street, watched by an armoured military vehicle.
Mr Al Hussein, the UN High Commissioner for Human Rights, said the victims had been "apparently cut down in a hail of gunfire".
The unrest came just days after security forces killed 20 Kurdish militants in clashes in nearby Diyarbakir, according to the Turkish military.
Three Turkish soldiers also died in a rebel attack, the military said.
The region has suffered its worst violence in two decades since a ceasefire between the Turkish government and the PKK collapsed in July.
The PKK, which has fought for autonomy for Turkey's Kurdish minority for decades, has been attacking security forces, while the army has been besieging Kurdish-dominated towns.
The Turkish Human Rights Foundation says at least 198 civilians, including 39 children, have died in areas under curfew since August.
Since July, more than 200 Turkish soldiers and policemen have been killed in attacks by the PKK, which is listed as a terrorist organisation by Turkey, the EU and US.
The European Union has called for an immediate ceasefire.

From 21 June, EDF customers will see electricity prices increase by 9% and gas prices go up by 5.5%.
The supplier previously increased electricity prices by 8.4% on 1 March, although it cut gas prices in January.
The rise was immediately criticised by the energy regulator, Ofgem, which said it was hard to justify.
The average dual-fuel customer will pay an extra Â£78 a year, bringing the annual bill to Â£1,160, EDF said.
Combining both increases will mean that 1.5 million EDF customers will pay 18.1% more for their electricity this year.
However, taking into account the previous cut, gas prices will remain flat, the company said.
Most of its customers - 1.8 million - are now on fixed price deals, so will not be affected.
Ofgem was highly critical of the increases.
Dermot Nolan, its chief executive, said: "EDF's second price rise in four months, when there has not been a dramatic rise in wholesale energy prices since it last put up prices, is difficult to justify and is further evidence that the energy market is not working in all consumers' interests."
Commentators said EDF customers on standard tariffs could save more than Â£300 a year by switching to the best fixed-price deal.
"EDF has hit millions with a second shock price hike," said Archna Luthra, head of energy at MoneySavingExpert.
"If you haven't compared and switched already, see this as a final red alarm bell to take action - what are you waiting for?"
EDF said it would be contacting all customers on standard variable tariffs to review their situation.
The company is also introducing a new three-year fixed deal.
"I know that price rises are never welcome, but the industry is facing significant cost increases," said EDF's chief executive, Vincent de Rivaz.
"To be a sustainable and responsible business, we aim to make a fair margin in supplying customers. This fair margin allows us to invest for the long term, in particular in good service, innovation and smart metering."
Five of the big six suppliers have announced price rises this year:
The top UN human rights official has urged Turkey to investigate an apparent shooting by security forces of unarmed civilians in the city of Cizre.
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Energy firm EDF is raising the price of electricity for the second time this year for customers on standard tariffs.