Officers were responding to an alleged shoplifting incident last month when the video was recorded.
Police officers can be seen shouting at the family to get out of their vehicle before threatening them.
The parents say they did not realise their four-year-old had taken a $1 (£0.79) Barbie doll from a store.
Mayor Kate Gallego said the officers' actions were "completely inappropriate and clearly unprofessional".Ms Gallego said in a statement: "There is no situation in which this behaviour is ever close to acceptable. As a mother myself, seeing these children placed in such a terrifying situation is beyond upsetting.
"I am deeply sorry for what this family went through and I apologise to our community."
She said that the city was speeding up the implementation of body-worn cameras. A community meeting about the incident will also be held on Tuesday.
In the video, Iesha Harper can be seen emerging from the car with her two young children. The children are handed to a bystander and Ms Harper is arrested.
The footage also shows another man, Dravon Ames, being kicked in the legs as he is handcuffed by an officer.
The couple are preparing to sue the city for $10 million over the incident.
Rapper Jay Z's Roc Nation company has offered the pair legal support.
Roc Nation Managing Director of Philanthropy Dania Diaz said in a statement: "We are calling for the immediate termination of the police officers in question. We are committed to supporting the family to ensure justice is served."
Ms Harper, who is pregnant, told CNN: "I really thought he was going to shoot me in front of the kids."
She said that she gave her two children to a bystander as she "didn't trust the police".
Phoenix police chief Jeri Williams said on local news that she was "sorry this incident happened" and that it was being investigated.
The officers involved have been assigned desk duty while the investigation takes place.