Smith was holding Rebello in a headlock and pointing a gun at her head before he turned his gun at the officer, Azzata said, prompting the shooting.
"He kept saying, 'I'm going to kill her,' and then he pointed the gun at the police officer," Azzata said.
A loaded 9 mm handgun with a serial number scratched off was found at the scene, police said.
Nassau County Police Commissioner Thomas Dale said he had traveled to Rebello's Tarrytown, N.Y., home to explain to Rebello's parents what happened.
"I felt obligated as a police commissioner and as a parent to inform them as soon as all the forensic results were completed," Dale said.
The veteran police officer, who was not identified, has about 12 years of experience on the Nassau County police force and previously spent several years as a New York City police officer, Dale said.
The officer is currently out on sick leave. He will be the focus of an internal police investigation once the criminal investigation is completed, which is standard police procedure in any officer-involved shooting, the commissioner said.
The shooting came just days before the school's commencement ceremonies, which are scheduled for Sunday.
A university spokeswoman said students will be handed white ribbons to wear in memory of Rebello. The shooting, which took place just steps from campus, has cast a pall over the university community as it geared up for commencement.
Earlier Saturday, police announced that Smith, 30, had been wanted on a parole violation related to a first-degree robbery conviction. A warrant was issued for Smith on April 25 for absconding from parole, police said.
Smith had what police described as "an extensive criminal history," which included arrests for robbery in the first degree in 1999, promoting prison contraband in the second degree in 2000, robbery in the first degree in 2003, assault in the second degree in 2003 and robbery in the second degree in 2003.
Rebello was in the two-story home in Uniondale, N.Y., with her twin sister Jessica, a third woman and a man when Smith, wearing a ski mask, walked into the house through an open front door, Azzata said.
The door was left open after someone had moved a car that was blocking a driveway, Azzata said.
When Smith entered, he demanded valuables and was told they were upstairs, Azzata said.
Smith, apparently unsatisfied with the valuables upstairs, asked if any of the four had a bank account and could withdraw money, Azzata said. The intruder then allowed the unidentified woman to leave and collect money from an ATM, telling her she had only eight minutes to come back with cash before he killed one of her friends, Azzata said.
The woman left for the bank and called 911, according to Azzata.
Minutes later, two police officers arrived at the home and found Rebello's twin sister Jessica running out of the front door and the male guest hiding behind a couch on the first floor, Azzata said.
One of the officers entered the home and encountered Smith holding onto Rebello in a headlock, coming down the stairs, Azzata said. Smith pulled Rebello closer and started moving backward toward a rear door of the house, pointing the gun at her head before eventually threatening the officer, Azzata said.
The Rev. Osvaldo Franklin, who gave Rebello and her twin their first communions, on Saturday night told The Associated Press their mother, Nella, couldn't even speak to him earlier in the day.
"She was so devastated," said Franklin. "She's just crying. We have to pray for Andrea, to pray for Jessica because she needs help."
Franklin said a funeral is scheduled for Wednesday at Teresa of Avila Church in Sleepy Hollow, N.Y., and will be in Portuguese.
"The family's a very good family, they have very good values," he said. "They are a very good, very devoted family."