Senior officers said they were shocked by the level of violence shown by the man towards his 44-year-old victim.
The terrified woman was walking by the Savoy Centre, in Sauchiehall Street, when her ordeal began.
As she walked by the shopping centre, she was approached from behind.
He punched her in the face before trying to prise her bag from her hands. However, the brave woman held on and the robber fled.
Police said the robbery took place at 11.30am on Tuesday.
Chief Inspector Alan Porte said the area would have been "very busy" at the time of the incident and urged witnesses to come forward.
Officers believe shoppers and city centre workers may hold vital clues in finding the culprit.
Mr Porte, area commander for Glasgow city centre, said: "Sauchiehall Street would have been very busy at this time of day.
"Someone must have seen something and I would urge any witnesses to contact officers.
"Thankfully, the woman was not seriously injured, but it is vital we find the man responsible.
"I'd like anyone who was in the vicinity of Sauchiehall Street or the Savoy Centre, at the time of the incident, to get in touch."
A trawl of CCTV is also under way in a bid to find out more about the attack.
The suspect is described as male, in his early 20s, around 5ft 8in tall, and had a gaunt appearance.
Meanwhile, detectives have stepped up their hunt for two sick thugs who attacked and robbed a woman in a wheelchair.
The 47-year-old was attacked mid-afternoon by two men who stole her handbag.
Officers said the terrifying incident took place in Waverley Park, Kirkintilloch, on Sunday at 3.45pm.
Detective Constable John Johnstone, said: "I believe that the park was busy last Sunday afternoon with families out enjoying the good weather and dog walkers."