"I just thank god that everything is being revealed — I know that he loved my child," DuBose's mother, Audrey, told reporters.
Tensing, who is white, is accused of fatally shooting DuBose after pulling him over because his vehicle was missing a front licence plate.
"He purposefully killed him," prosecutor Joe Deters said Wednesday. "He should never have been a police officer."
Deters said during a news conference Wednesday that he thinks Tensing lost his temper.
The prosecutor's office released body-camera video footage Wednesday of the July 19 shooting near the University of Cincinnati. Deters told reporters he was "shocked" after viewing the recording.
The footage Deters played for reporters shows Tensing and DuBose talking. Tensing asks for DuBose's licence, but DuBose says he doesn't have it on him. A gunshot is heard, then Tensing is seen running after DuBose's slowly rolling car, not being dragged by the car as the officer had reported.
"I've been doing this for 30 years. This is the most asinine act I've ever seen a police officer make, totally unwarranted," Deters told reporters after meeting with DuBose's family.
Deters also said Tensing failed to issue simple, non-violent commands.
"He wasn't dealing with someone who was wanted for murder," Deters said. "He was dealing with someone who didn't have a front licence plate. This is, in the vernacular, a pretty chicken crap stop."
The family saw the video on Wednesday for the first time and met with Cincinnati Mayor John Cranley.
DuBose's family has hired attorney Mark O'Mara, who represented George Zimmerman, the Florida man who was acquitted in the 2012 shooting death of black teenager Trayvon Martin.
"Seeing that video let me know that my son did absolutely nothing, not nothing, nothing to provoke this man," Audrey DuBose said. "I think the person should have been locked up on day 1. He should have never been released. If they saw the film, why was he still walking the streets?"
The victim's sister, Terina DuBose Allen, credited the video with bringing about justice for her brother.
"If it were not for that video camera, Sam would be no different than all of the other incidents," she said.
DuBose's death comes amid several months of national scrutiny of police and their interactions with black citizens.
Tensing's attorney had said earlier he wouldn't be surprised if his client were indicted given the political climate.
Officer turns himself in
Police said Tensing, who also was charged with voluntary manslaughter, surrendered at about 2 p.m. ET.
Tensing's attorney, Stew Mathews, told the Fox 19 television station in Cincinnati that his client was being "thrown under the bus" by the prosecutor and the school.
Tensing has more than five years of experience in law enforcement and has worked as a University of Cincinnati police officer since April 2014, Goodrich said.
An Associated Press review of his personnel file didn't indicate any problems in that time.
If convicted, Tensing could face life in prison.
Police department review called
Deters said the university should not have its own police force and that the city police, who he said were better trained, should take over campus security.
The mayor said Wednesday there will be an independent review of the University of Cincinnati Police Department.
And the university said it will review its policing strategy in the wake of the shooting.
Earlier Tuesday, University of Cincinnati president Santa J. Ono offered condolences to DuBose's family and loved ones.
"Our hearts grieve for his loss," Ono said. "We also know that police officers risk their lives every day, and when their efforts to protect themselves and our community result in a death, it is a tragedy. No matter the circumstances, it is a time of unimaginable sadness for all involved."
City leaders pledged Tuesday to make sure the case is given a thorough, transparent review.
"This is a serious situation, and I will do everything necessary to ensure the investigation is handled as such," city manager Harry Black said in a statement.