Fire and rescue spokesman Jason Evans said the person who died had been found inside the building and the injured had been taken to hospitals, two of them in a critical condition.
The person who died was a woman, according to local media reports.
Emergency services were called to the Elan City Lights apartment building just before 2pm local time after the National Weather Service issued a severe thunderstorm warning for the Dallas/Fort Worth area.
Resident Kevin Collins, who was inside one of the apartments when the crane fell on the building, said: "I'm highly grateful, I could've been dead.
"The man next door, I believe, he's sustained, like, critical injuries to his neck. He could not move, he could not stand. Two people had to grab him and take him out, and the shockwave, really, just shocked the whole building. Everyone is just shook up."
A woman identified as Abbey Kearney told CNN that she and her husband saw the crane come down on the building just as extremely high winds kicked up in the area.
"It just sliced through the building... like a hot knife through butter," she said.
The crane broke into several pieces that fell into different portions of the apartment building, located across the street from a large construction site, Mr Evans said.
He said did not know whether the crane was in operation at the time.
While the exact cause of the accident is not yet known, Mr Evans said there was a "strong possibility that the wind did play some role in the collapse of the crane itself".
Across Dallas on Sunday the storm felled trees and knocked out power lines.