One person believed to be in critical condition was sent to the hospital, while two others have non life-threatening injuries, Tiffany Ngalula, a public information officer for the Glendale Police Department said in a livestreamed media briefing.
Other videos taken by those at Westgate showed people scrambling for safety, with sounds of booming gunfire and screaming babies in the background.
Employees at businesses in the Westgate area have been asked to shelter in place while police complete a secondary search.
"The suspect in custody is 20-year-old Armando Hernandez Jr".
Police have not released the names of the victims or the suspect.
The video surfaced on social media in the minutes after the shooting was first reported about 7:40 local time and included an interaction with a person who appeared to be shot in a parking lot asking for the shooter to stop.
State Sen. Martin Quezada tweeted that he witnessed a shooter in the area armed with an assault weapon.
Suspect in Custody After 3 People Shot at Arizona Shopping Center
At least three people were shot on Wednesday night at the Westgate Entertainment District shopping complex in a suburb of Phoenix, Arizona.
"I just witnessed an armed terrorist with an AR-15 shoot up Westgate", he wrote.
Gov. Doug Ducey tweeted that his office was monitoring the situation closely and the state is ready to assist as necessary. He said he saw "multiple victims".
Quezada later added: 'I saw 2 victims with my own eyes. "As we learn additional details, the state is here to continue to offer its full support to the victims and to the community". Being told not to say anything else about details until I speak to police. "Not sure how many others".
However, the person on camera identified himself as Armando Junior Hernandez and said in part, "I'm gonna be the shooter of Westgate".
The shooting comes as malls, restaurants and shops in Arizona are reopening.
This is a developing story.
United Kingdom to mass produce COVID-19 vaccine at its new £38Mn center
If the trials are successful, they will make 30 million vaccine doses for Britain by September, and 100 million doses in total. A doctor working on the coronavirus vaccine has told LBC the jab is unlikely to be ready by September .