Police identified the suspect as Arthur D. Harmon. They identified a man who died hours after the Wednesday morning shooting as 48-year-old Steve Singer.
Police say a 43-year-old man was listed in critical condition along with a 32-year-old woman.
Police didn't release the names of the wounded, but a Phoenix law firm, Osborn Maledon, said one of its lawyers, Mark Hummels, was among the wounded.
The firm said he "was representing a client in a mediation" when he was shot.
As police searched for the shooter, SWAT teams and two armored vehicles surrounded his house about 7 miles from the shooting scene. Police served a search warrant to enter the home.
For a time, officers, believing the shooter was inside, used a megaphone to ask him to surrender.
The gunfire at the office complex prompted terrified workers to lock the doors to their offices and hide far from the windows. SWAT officers searched the building.
"Everyone was just scared, honestly, just scared," said Navika Sood, assistant director of nursing at First at Home Health Services who along with her co-workers locked the entrances to their office.
Sood said police evacuated the office about 30 minutes after she first heard the popping noises.
The shooting took place on the same day that hearings on legislation to address gun violence were convened in Washington, with former Rep. Gabrielle Giffords testifying for stricter gun controls.
A gunman shot Giffords in the head during a shooting rampage in Tucson in January 2011.
Around 10:30 a.m., the gunman arrived at the office building and got into a dispute with someone, a conflict that escalated to the point where he drew a gun and shot three people, Thompson said.
Vanessa Brogan, who works in sales support at an insurance business in the three-story complex, said she heard a loud bang that she thought at first was from somebody working in or near the building.
She said others at the business thought they heard multiple loud noises. She said people locked themselves in offices until authorities evacuated the complex that houses insurance, medical and law offices.
Becky Neher, who works for a title company in the building, said the two gunshots she heard sounded like two pieces of metal banging against each other.
Watching from her second-story office, she saw people leaving the building.
"Someone yelled, 'We have a shooter,'" she said. She saw two victims lying on the ground outside the back side of the building. She said health care workers who have offices in the complex came out to help.
Don Jaksa, a software consultant who works in the building, said he was listening to the radio when he suddenly heard "two pops." He said he didn't think they were gunshots.
"My co-worker goes to the range all the time," he said. "He identified it as gunfire."
His co-worker then locked the door. After five minutes, they left and ran into police and someone carrying a stretcher. The police escorted them back to their office and told them to lock the door again.
They were eventually evacuated, and as he sat on a rock outside the complex, his wife called to make sure he was OK after seeing the shooting on the news.
Workers were later allowed to leave the building. Two hugged each other when they got outside.
"You don't expect this when you come to work," worker Lindsa Rincon said.
Associated Press Writers Paul Davenport, Felicia Fonseca and Terry Tang contributed to this report.