The finale aired after FBI agents arrested Durst on an extradition warrant out of Los Angeles just before 4:30 p.m. Saturday at the JW Marriott hotel in New Orleans, where he had registered under an alias and paid for the room in cash. Fake documents were found in his possession.
The FBI believed he would flee, possibly to Cuba, as inaugural flights to the country began Saturday from New Orleans.
"We are relieved and also grateful to everyone who assisted in the arrest of Robert Durst. We hope he will finally be held accountable for all he has done," said Durst's brother, Douglas Durst.
In the finale, filmmakers asked Durst about similarities in handwriting in a letter he wrote and another linked to one of the killings to which he has been linked, The Associated Press reported, before he wore his microphone into the bathroom.
What followed was a bizarre rambling in which Durst said, apparently to himself, "There it is. You're caught" and "What the hell did I do? Killed them all of course."
The 71-year-old faces charges for the murder of his one-time friend Susan Berman, whose body was found in her Beverly Hills home in 2000 as New York authorities prepared to question her about the 1982 disappearance of Durst's wife, Kathie.
After Berman's death, Durst moved to Texas, where he lived as a woman before being acquitted in the 2001 dismemberment death of his Galveston neighbor, Morris Black. Durst said that killing was in self-defense.
In this Aug. 15, 2014 file photo, New York City real estate heir Robert Durst leaves a Houston courtroom.
Defense lawyer Chip Lewis, who successfully defended Durst in the Texas killing, said his client will waive extradition and be transported to Los Angeles to face the charges.
"He's maintained his innocence for years," Lewis said. "Nothing has changed."
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The HBO true-crime series "The Jinx" explores Durst's 2003 murder trial in Galveston and the unsolved, suspected killings of Berman and Kathie Durst.
The series led to a never-before-seen letter discovered by Berman's stepson Sareb Kaufman. Kaufman told ABC News he believes Durst sent the letter to Berman a year before her death, and claims it carries similarities to an anonymous note sent to police by Berman's killer. That handwritten letter was recently handed over to authorities in California, who reopened their investigation into Berman's death.
Durst denies killing Berman, but police have long believed he had a motive, allegedly wanting to silence Berman about the 1982 disappearance of his first wife, Kathleen.
Former judge Susan Criss, who presided over Durst's Galveston trial, says she was "thrilled" to hear that he had been arrested.
"He was exonerated because of the fact that the prosecution did not do their job. I suspect a couple of jurors might have had ulterior motives," Criss said.
"It's amazing how many times the ball has been dropped when he has been suspected of murder, but I'm hopeful for this one to stick so that the families can have closure," Criss added.
Lewis, the attorney, said the arrest was orchestrated by Hollywood to come before the final episode, which airs Sunday.
"No doubt," he said. "It's all about Hollywood now."
Lewis said he was familiar with the Berman killing and wasn't surprised by the arrest because of the number of emails and calls he got after last week's episode aired. He said new evidence touted by producers, however, was something he was already familiar with.
"I know all about this case," Lewis said. "I have no doubt we will present a most compelling defense."
Filmmaker Andrew Jarecki told a Hollywood version of Durst's story in the 2010 film that starred Ryan Gosling, "All Good Things."
A week before the release of that film, Durst called Jarecki saying he wanted to see it, and eventually agreed to be interviewed by Jarecki. That footage led to the documentary series.
Jarecki said he has come to a "firm conclusion" about Durst's guilt or innocence.
HBO distributed the first two episodes in advance, making news with Durst's admission that he lied to investigators about what he did on the night of his wife's disappearance. The other episodes were kept under wraps to maintain suspense as they aired each week.
ABC News and the Associated Press contributed to this report.