Albert Anthony Smith McDonald, 58, was found with head injuries on Cumbernauld Road, Riddrie, at 21:25 on Monday. He died on his way to hospital.
Officers said the car driver is assisting with their inquiries.
Police said Mr McDonald's family were "completely devastated" by his death and appealed for information.
Senior investigating officer Det Ch Insp Laura McLuckie said earlier: "We are following up several lines of inquiry, including speaking to local residents and accessing CCTV from around Cumbernauld Road and the surrounding area."
She added that "Mr McDonald had significant injuries" when he was found and the investigating team was "trying to establish just how he sustained them".
She said officers were trying to piece together his last known movements.
"Understandably his family and friends are completely devastated, and specialist officers are providing them with support at this tragic time," she said.
"It is imperative we find who is responsible for Albert's death.
"I would ask anyone who was in the vicinity of Cumbernauld Road around 21:30 or earlier, and who may have seen anything suspicious to come forward to police."
Det Ch Insp McLuckie added: "You may think that the information you have is insignificant however it may be the piece of information we are looking for."

Williams, 42, was forced to qualify after he was defeated by Mark Selby on Sunday in the final of the China Open.
He will be one of 14 Welsh players who will have to win three qualifying rounds at Ponds Forge in Sheffield to secure a place in the first round.
Ryan Day is the only Welshman to be guaranteed a place in the competition.
The World Championship starts at Sheffield's Crucible Theatre on Saturday, 15 April.
15-year-old schoolboy Jackson Page and amateur Tyler Rees, 18, will play in the qualifying rounds after receiving an invitation to compete.
Page, of Ebbw Vale, will take on world number 75 Martin O'Donnell on Thursday while Rees will play against world number 44 Xiao Guodong of China on Wednesday.
Elsewhere, Matthew Stevens - twice a runner up at the Crucible - will face off against fellow Welshman Lee Walker on Wednesday.
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PC Ed Barker, of Greater Manchester Police (GMP), stopped the man falling 40ft (12m) from Rochdale Town Hall.
Seeing the man in distress, PC Barker ran up the building's fire escape steps before catching him just as he tried to throw himself to the ground.
PC Barker has been awarded the force's John Egerton Trophy.
Chief Constable Ian Hopkins presented the trophy, which is awarded annually to a GMP officer who has performed an exceptional act of valour.
The award is named after PC John Egerton, who was killed in the line of duty on 11 March 1982, aged 20.
PC Barker is also due to meet Prime Minister David Cameron during the Police Bravery Awards on 14 July.
Mr Hopkins said his officer showed "incredible bravery" during the incident, which happened in January 2015.
"There was a genuine risk of him being carried over the edge of the snow-covered fire escape, which could have had tragic consequences for both men," he said.
"His cool head and quick thinking saved this young man from certain death and I am delighted that he is now safe and well."
PC Barker said the 21-year-old man's friends had tried to help him but were "losing their strength" when he arrived.
"I managed to get hold of him around the neck and shoulders because the rest of his body was hanging off the edge of the fire escape," he said.
"I was so relieved when I saw the blue lights arriving to help take him to hospital.
"It's nice to get some recognition and appreciation, but I was helped by his friends and my colleagues."
Police investigating the murder of a man who was found seriously injured on a Glasgow street have traced the driver of a car seen near where he was found.
[NEXT_CONCEPT]
Two-time world champion Mark Williams will face China's world number 84 Zhao Xintong in the World Championship first qualifying round on Saturday.
[NEXT_CONCEPT]
A police officer who saved the life of a suicidal man by grabbing him by the neck as he tried to jump from a snow-covered roof has won a bravery award.