Flames ripped through the upper floors of the eight-storey building in York Street and fire crews were called at around 11.20pm on Sunday.
Firefighters battled through the night to bring the fire under control and yesterday morning they began conducting the salvage operation in affected flats.
The cause of the blaze remains unknown, but the Incident Research and Investigation Section of Strathclyde Fire and Rescue is investigating.
Dozens of residents were evacuated from the building.
York Street was closed between Argyle Street and Broomielaw, and early morning commuters struggled to negotiate the cordon to reach their offices.
One 39-year-old resident, who was evacuated from the top floor flat he shares with his wife, described the fire as "spectacular".
He said: "I wasn't sure what was going on because I couldn't see any flames or smoke.
"I didn't realise there was such an extensive fire until we got outside and looked up and saw the flames."
People living nearby saw flames leaping from the windows of the penthouse at the top of the building and plumes of smoke billowing from the roof.
This morning, the top of the building was burnt out and blackened and the windows of the top floor apartments were blown out.
There was debris on the street, which appeared to have fallen from the metal guttering that had become warped and blackened.
More than 40 firefighters from Cowcaddens, Calton and Maryhill Community Fire Stations attended the blaze after the public reported that the building was on fire.
Firefighters arrived to find the sixth, seventh and eighth floors engulfed in flames.
A further five fire tenders and additional specialist appliances were immediately called to the scene, while firefighters in breathing apparatus searched the properties.
One woman was rescued from the building after suffering slight smoke inhalation.
Assistant chief officer Lewis Ramsay, the incident commander, said: "Initial crews worked tirelessly in extreme conditions to extinguish this fire and minimise smoke and water damage."