SAN JOSE — A San Jose police sergeant was recovering Thursday after a bullet grazed his head during a close-range gunfight Wednesday with a carjacking suspect who later died when shot and struck by a police SUV in downtown San Jose, authorities said.

Police gave the update one day after a sprawling, nearly two-hour chase that stretched from Hollister to San Jose and involved multiple law enforcement agencies. San Jose police Chief Paul Joseph said at an afternoon news conference that the sergeant, who has 13 years of service with the department, suffered a skull fracture.

The sergeant was released from the hospital Thursday afternoon. Police identified him as Gerardo Silva.

Joseph said that near the end of the incident, the suspect collided with a stopped vehicle near Notre Dame Avenue and West Julian Street. As Silva was in pursuit, the suspect got out of the car and “charged” at him, Joseph said, immediately firing.

Silva was struck in the head as he arrived, Joseph said, but still managed to get out of his vehicle and return fire with another officer. The two circled the police vehicle, exchanging gunfire. The suspect briefly tried to enter the police vehicle, then fled on foot before he was shot and killed seconds later by approaching officers.

Silva was initially listed in critical condition and was rushed to a hospital.

“This was a battle for that sergeant’s life, unfolding in the middle of our city, in broad daylight, with members of the public in the crossfire. … Under fire, wounded and under extreme stress, his training and resolve kicked in.”

Steve Slack, president of the San Jose Police Officers’ Association, said in a statement that officers displayed “incredible bravery” as they ran toward gunfire and “ultimately eliminated the threat.” He said Silva remained in good spirits and that the union was supporting him and his family.

San Jose Vice Mayor Pam Foley said she could not imagine “the courage it takes to run toward a situation like that and not away from it.”

“To our sergeant who is recovering, we thank you for your dedication and commitment to protecting our residents, and we pray to you have a speedy recovery,” she said.

City Manager Jennifer Maguire said she visited Silva Wednesday night.

“The sergeant was in very good spirits and extraordinarily positive, given the tragic situation,” Maguire said. “However, I’m not surprised by this, as his colleagues could not say enough about him and the impact he has made on this community,”

“Most of all, I’m thankful that our sergeant survived an unimaginable injury,” Joseph said.

Staff writer Jason Green contributed to this report.