Brown, 52, died at 12:56 a.m. PDT, after a four-member team injected a lethal one-drug cocktail in the execution chamber of the Washington State Penitentiary.
The father, brother and two sisters of his victim, Holly Washa, 21, witnessed the execution, as did King County prosecutor Dan Satterberg.
Brown complained that criminals who had killed many more people, such as Green River killer Gary Ridgway, were serving life sentences.
"I only killed one victim," media witness Liz Rocca of KOMO-TV in Seattle, quoted Brown as saying.
Media witnesses said Brown did not apologize to the family of the victim, but said he understood their hatred for him. He said he forgave that hatred, and hoped the execution would give them closure.
After his comments, witnesses said Brown, who was lying on his back strapped to a gurney, looked up at the tubes sticking out of the wall and connected to his body. Death seemed to come quickly when the drugs were administered immediately after the statement, witnesses said.
Brown's attorney and members of his family were not present at the execution, though he spoke with them by phone on Thursday.
The U.S. Supreme Court, 9th U.S. Circuit Court of Appeals, and the state Supreme Court on Thursday all rejected efforts to stay the execution. Gov. Chris Gregoire rejected his plea for clemency on Wednesday.
It was Washington's first execution since 2001, and Brown had been on death row for 16 years.
"This is a somber end to the long journey Holly Washa's friends and family have experienced since losing their loved one in such a tragic way nearly 20 years ago," said Attorney General Rob McKenna.
Brown had argued that his mental illness was not adequately considered during his sentencing and that it should bar his execution. According to court records, he suffered from bipolar disorder.
Brown confessed to killing the 21-year-old Washa during an interrogation in California for an alleged assault on a woman there. He later led authorities to Washa's battered body, which was inside the trunk of a car.
He met Washa near Sea-Tac airport in Washington when he helpfully pointed to Washa's rear tire, indicating a problem. When she stopped to check it out, he carjacked her at knifepoint.
For the next 36 hours, Brown robbed, raped and tortured Washa, before stabbing and strangling her.
"His demeanor - that's what struck us. And his lack of remorse," said Lt. Al Franz of the Palm Springs police department, one of the investigators who first interviewed Brown in California.
"This is a violent individual, and he was just very, very calm while he was telling his story," Franz said. "The lack of remorse was pretty incredible to me. The way he spoke about his victims, they weren't people to him."
Brown, who is from San Jose, Calif., had a history of violence against women, including a 1977 conviction in California for assaulting a woman with a knife at a shopping center. He also served 7½ years - the minimum sentence - for assaulting another woman in Oregon in 1984.