Video shared on social media showed flames engulfing the lower part of the building and thick black smoke billowing from the upper floors.
Many of the casualties are said to be foreign workers who lived there, including a number of Indian nationals.
Indian Prime Minister Narendra Modi sent his condolences to the victims and their families.
"The fire mishap in Kuwait City is saddening," he said on X.
"My thoughts are with all those who have lost their near and dear ones. I pray that the injured recover at the earliest."
He said the Indian embassy was monitoring the situation and working with the authorities on the ground.
Kuwaiti Deputy Prime Minister Sheikh Fahad Yusuf al-Sabah accused property owners of greed and said violations of building standards had led to the tragedy.
"Unfortunately the greed of the property owners is what led to this," Sheikh al-Sabah, who is also acting interior minister, told Reuters news agency.
"They violate regulations and this is the result of the violations."
Interior Ministry spokesman Maj-Gen Eid al-Oweihan told state TV that the fire was reported at 06:00 local time (03:00 GMT) on Wednesday. It was now under control.
An eyewitness, Manikandan from the Indian state of Tamil Nadu, told BBC Tamil that many of the workers had been on night shifts.
"Some of those who returned to that apartment early in the morning were cooking food after coming back from work," he said.
"Once the fire erupted, it spread rapidly. People living in the building were not able to control the fire."
He said many of the building's residents were from Tamil Nadu and another southern Indian state, Kerala.
Local media reports say the building housed 196 workers and there are suggestions that it may have been overcrowded.
A senior police officer told state TV that there were a "large number" of people in the building at the time of the fire.
"Dozens were rescued, but unfortunately there were many deaths as a result of inhaling smoke from the fire," he said, adding that warnings were often issued about overcrowding in this type of accommodation.
No details have so far been given officially about the workers' countries of origin or the nature of their employment.
But rescue worker Shamsudheen Kanneth told the BBC that of the 49 dead, 30 had been identified, most of whom were Indian nationals.
There were also Pakistanis, Bangladeshis, Filipinos and Nepalis among them.
However, some of the bodies had been burned beyond recognition, Mr Kanneth added, and DNA tests would be required.
Two-thirds of the Kuwaiti population is made up of foreign workers and the country is highly dependent on migrant labour, especially in the construction and domestic sectors.
Human rights groups have regularly raised concern over their living conditions.
By Robert Greenall