Nick Pierpoint said he was shocked by the incident which happened at Golf Madness in Corn Market at about 6.40pm on Monday evening.
He said the man ran from the building with his coat on fire before abandoning an incendiary device in the street and fleeing through crowds of shoppers.
Police sealed off part of city centre in the wake of the attack, which came just days after a device containing flammable liquid exploded outside a restaurant in Cathedral Quarter.
They believe the man may have suffered burns to his face, neck and hands after the device went off prematurely.
"The whole thing happened so quickly, probably in the space of 10 or 15 seconds," Mr Pierpoint told UTV on Tuesday morning.
"We were in the shop and it was reasonably busy. A guy came in and he looked a bit agitated, as though he was flicking at something like a cigarette, then within the next five or six seconds his coat literally burst into flames, and he ran out of the shop.
"I could see the reflection of the flames outside and I thought, because he was burning so badly, he was lying on the street - so I went out to try and help him.
"But it turned out it wasn't him, he had went on and it was the device burning in the street."
None of the roughly 15 people who were inside the pop-up golf store were injured when the device went off inside the man's jacket.
A large flame ignited inside the male's jacket and this engulfed his head and shoulders and he attempted to put the flames out with his hands.
Chief Superintendent Alan McCrum
The man, who was wearing a beanie hat, was seen desperately trying to beat out the flames.
Chief Superintendent Alan McCrum said: "We believe that this man may require treatment for burns injuries to his neck, face or hands.
"The suspect involved in this incident is believed to be approximately 6ft tall, of plump build and was wearing a dark coloured waterproof coat which was hip length with a drawstring at the bottom and dark jeans or tracksuit bottoms.
"The male ran from the shop in the direction of Ann Street and the device was kicked out of the door of the shop and ended up in Cornmarket."
Mr Pierpoint added that he was not a young man but was probably aged in his 40s, and said he would be shocked if this incident was not linked in some way to Friday's.
He continued: "I would guess he was in his early 40s, so it was not a young fellow. I don't want to put two and two together and get five but when you look at what's been going on recently in Belfast I'd be surprised if there wasn't some sort of connection."
Army bomb experts examined the device and removed it for further examination before the area was reopened after several hours.
Police have advised city centre businesses to check their premises for any suspicious objects after the latest terrorist incident linked to dissident republican activity in the city.
Thousands of people have been stopped at security checkpoints following an attempted car-bombing at the Victoria Square Shopping Centre, while officers are continuing to try to identify a man who left a bomb in a sports bag at Exchange Street West.
Police said the device fully exploded while up to 1,000 people were being evacuated from nearby pubs, clubs and restaurants on Friday night - no-one was injured.
Glyn Roberts, who represents the local business group NIIRTA, said traders will not be put off by this increase in attacks during the busy festive season.
He said: "I know that there is still an absolute determination amongst traders and shoppers to continue on and not be dragged back by people behind these attacks
Anyone who was in Corn Market at the time of Monday's attack and who may have noticed any suspicious activity is also asked to contact PSNI detectives at Musgrave.
Mr McCrum added: "Police are doing everything to protect Belfast and other places to enable people to go about their business peacefully.
"We would appeal to any care provider who may already have been approached or who is approached today, by a male in his 40s with burns to the neck or face to contact detectives directly at Musgrave station on 0845 600 8000."