The man forced a security guard to hand over cash as the van made a delivery to Lambhill Post Office, in Balmore Road.
After threatening the terrified guard with a gun, the suspect, who escaped with an undisclosed sum of money, made off on foot.
He is described as being around 5ft 10in tall, was wearing dark clothing and had his face covered.
Police said no shots were fired before the culprit ran off on towards Knapvale Street, a quiet residential road off Balmore Road.
The robbery happened outside the busy Post Office at 12.10pm.
Police raced to the scene, in the north of the city, and immediately set up cordons.
Just minutes later, fire crews were called to a car which had been set alight, at King's Place, less than half a mile from the scene.
Detectives were unable to confirm whether the fire-damaged silver Volvo S40 was linked to the post office van raid.
One resident, who lives near the post office, but did not wish to be named, said: "One of my neighbours told me she'd seen the guy with the gun at the van.
"The workers were all standing about outside, they looked in a terrible state, very shocked.
"I'd heard there was an old lady inside the post office at the time.
The van stops at the post office several times a day, so I wouldn't be surprised if this guy has been watching the van, or following it.
"It's scary to think a guy was running about with a gun here in the middle of the day."
Detective Inspector Stephen Healy, from Maryhill CID, said: "This was a terrifying experience for the security guard, who has been left very upset.
"I would like to speak to anyone who may have seen a silver Volvo S40 in the vicinity at the time.
"Or a man fitting the description of the suspect prior to the incident.
"This was a very brazen act, to target a security guard in the middle of a very busy road in broad daylight, and it is very important we trace the dangerous individual who carried out this despicable attack."
A spokeswoman for Strathclyde Fire and Rescue, said: "Fire crews were called just after 12.15pm to attend a vehicle fire on King's Place, in Glasgow.
"Crews extinguished a car. No-one was reported as being in or with the vehicle at the time."
Anyone with any information is asked to contact officers at Maryhill CID on 0141 532 3700.