The victims most likely suffocated in the basement of the building, officials said, and authorities have opened a criminal investigation.
Local media said the fire started in the basement of the three-floor building, which was turned into a hostel several weeks ago, and that it required repair work.
"Thirteen dead people have been discovered, and their identities are being established," the Almaty city emergency situations department said.
"Preliminarily, they died of carbon monoxide poisoning."
The fire broke out in the centre of the former capital at around 5:30 am (2330 GMT).
Firefighters arrived seven minutes later to find the basement of the residential building burning, according to the statement.
The fire was extinguished within around an hour.
Emergency services said that 72 people were staying at the hostel at the time of the fire, and 59 were safely evacuated.
Authorities said the building was converted into a hostel a month and a half ago. The cause of the incident is under investigation.
Russian media reported two Russian citizens were among the dead.
Kazakh media said the ground and basement floors of the hostel had been turned into dorms, with some publishing images of a cramped room without windows with at least five wooden bunk beds.
A lax approach to safety regulations in Kazakhstan is widespread, often leading to deadly situations.
