The cygnets and their mother had recently become a familiar sight around Lisnarick, outside Irvinestown.
Pat Masterson, who lives in Lisnarick, photographed the swans shortly before they were killed on Tuesday evening.
"When I saw their mother walking them along the road, all in line, I stopped to take a picture," said Ms Masterson.
"They've  visited our farm every day over the last few weeks so I've photographed them before, but I thought that the way the wee cygnets were following their mother was just so lovely I had to stop and watch.
"I still can't believe that they were run over so soon after."
Enniskillen-based vet Kevin Leonard, who happened to be passing moments after the birds were killed, said he was shocked by what he saw.
"The five of them were just lying in a row behind the mother, who was injured as well.
"Four of them were dead and the fifth was so badly mangled I had to put it down with an injection.
"The sixth one only had minor injuries, so I gave it an anti-biotic cream and guided it into a nearby field alongside its mother.
"It was actually quite an upsetting sight. I can't believe whoever killed them just drove on."
Paul Hoy, who had also stopped when he saw the dead swans, assisted the vet in moving them to safety.
He said he was "disgusted" by what he saw.
"A girl who was there said she saw the car running them down and driving on.
"We're all wondering how this could have been an accident - it's a 30mph speed limit there and it wasn't dark so the swans were very visible.
"If you can't see six swans on the road, what are the chances you'll see a child?
As native wild birds, swans enjoy statutory protection under the Wildlife and Countryside Act 1981 and it is an offence to intentionally injure, take or kill a wild swan.
The act similarly protects the eggs and nests of swans.
It is an offence to take or possess the egg of a wild mute swan, or to damage or destroy the nest of a mute swan whilst in use or being built.
The RSPB said anyone who witnesses an offence against swans or other wildlife should contact their police Wildlife Liaison Officer through their local police station.

Fujimori said in a letter he intended to share thoughts and memories, happy and sad, while serving a 25-year prison sentence for human rights violations.
The social media accounts would be managed by a group of supporters.
The 75-year-old governed Peru for a decade before being impeached and fleeing the country in 2000.
Peru's Justice minister, Daniel Figallo, reacted angrily to Fujimori's announcement.
"People who commit violations and have their liberty taken from them have their rights limited. Otherwise, we turn prisons into hotels," he told reporters.
Mr Figallo reportedly asked the prison director to limit Fujimori's access to social media.
The former leader still has supporters in Peru and vowed to reveal his thoughts online.
"In this way, I'll be able to share my thoughts, memories and illusions,  sorrows and joys," his letter read.
Peru's President Ollanta Humala recently rejected Fujimori's request for a pardon on humanitarian grounds.
Earlier this week, the authorities rejected a bid by Fujimori's lawyer to have him  transferred to house arrest.
After being stripped of his powers by the country's Congress in 2000, the leader sought exile in Japan.
He eventually returned to Chile, where he was arrested in 2005.
After losing a lengthy legal battle, he was extradited to Peru in 2007 and convicted of human rights violations after a 15-month trial.
He was accused of authorising death-squad killings in two incidents known as La Cantuta and Barrios Altos, and the kidnapping of a journalist and a businessman.
Mr Fujimori repeatedly denied the charges, saying they were politically motivated.
Residents in a County Fermanagh village have expressed anger at the "heartless act" of a driver who killed five cygnets.
[NEXT_CONCEPT]
The former president of Peru, Alberto Fujimori, has said he will publish excerpts from his memoirs on the social media websites Twitter and Facebook.