The footage shows the youth, identified as Sarfraz Shah, apparently pleading for mercy after being surrounded by five troopers of the Pakistan Rangers at Benazir Bhutto Park in Boat Basin area of Karachi.
One of the troopers then fires two shots at Shah from point blank range with a G3 assault rifle.
Pakistan Rangers officials had earlier told the media that Shah was an "armed criminal" and was killed when he resisted efforts to capture him after a robbery.
The officials claimed security personnel had shot Shah after he fired at them.
However, the footage aired on all TV news channels clearly showed that Shah was unarmed when he was shot.
The footage shows Shah being grabbed by the hair and roughed up by a man clad in a white shirt who has a pistol in his hand.
Reports said this man was a policeman in plainclothes.
The man then hands over Shah to five paramilitary troopers, one of whom rams his G3 rifle against Shah's throat.
Shah is then seen pleading for mercy with folded hands. Shah approaches the trooper with the rifle and appears to be asking him to lower the weapon when another trooper pulls them apart.
The trooper with the G3 rifle then fires two shots at Shah, who shouts out in pain and crumples to the ground as the bullets hit him in the stomach.
One trooper, who appears to be older than the others, turns his face away in shock following the shooting.
Reports said Shah, who succumbed to his injuries in hospital, was the brother of a journalist.
The dead youth's relatives and friends staged a protest outside the Sindh Chief Minister's official residence with his body for over six hours.
Interior Minister Rehman Malik, who was in Karachi for an official visit, said he had ordered a magisterial inquiry into the incident.
Police officials said two of the troopers involved in the incident had been arrested and the three others would also be taken into custody.
"I have taken very serious notice of the matter. Even if the youth was involved in looting and handed over to the Rangers, they had no right to shoot him. I assure the people I will take very serious action and justice will be done," Malik told the media.
The paramilitary Frontier Corps was embroiled in a controversy after its personnel gunned down five suspected Russian nationals, including three women, in the southwestern city of Quetta on May 17.
Officials claimed the foreigners were armed and wearing suicide vests.
Footage shown on television showed the foreigners were unarmed when they were killed by the security personnel.
An autopsy of the foreigners revealed they died from "multiple bullet wounds", contradicting claims by officials that they died because of an explosion.
The footage on TV showed security personnel firing volley after volley of bullets at the foreigners as they lay on the ground near a check post made with sand bags.
One wounded woman waved her hand in the air before she died.
