He died at 8:45 a.m. PDT Tuesday after a long illness, Bragman said.
Conway's celebrated comedy career included acclaimed performances on such programs as Burnett's variety show and "McHale's Navy." Conway won four Emmys for his work on "Burnett" and two more for guest appearances on "Coach" and "30 Rock."
In a statement relayed to USA TODAY by her representative, Steve Sauer, Burnett said she was "heartbroken" to hear of Conway's death.
"He was one in a million, not only as a brilliant comedian but as a loving human being," she said. "I cherish the times we had together both on the screen and off. He’ll be in my heart forever.”
Other celebrities, especially fellow comedians, paid tribute on social media. "RIP to Tim Conway who was always always always always funny!" tweeted comedian Larry Wilmore.
Conway's daughter, Kelly, and five sons announced his death in a news release sent to USA TODAY by Kelly Conway's publicist, Roger Neal.
"The love he gave us, and the laughter he gave the world will never be replaced but will be remembered forever," Kelly Conway said in a statement. "He is at peace now, but I will miss him every second of every day until we meet again in heaven. We knew he would have to leave us someday, but that day came too soon."
She said her father used to say, when he would hear the song “Angel Flying Too Close to the Ground,” that he wanted that song played at his funeral.
Neal, who was Conway's publicist for five years, added in a statement to USA TODAY that Conway was a joy to work with.
"He was offscreen what the world saw onscreen," Neal said. "We had so many laughs. … What a pure joy to have had the privilege of working with one of the true masters of comedy. … There will never be another one like him."
According to court documents in Los Angeles, Conway had declined into dementia in the last months of his life, leading to a legal struggle last year between Kelly Conway and his wife, Charlene Conway, over guardianship and which one would decide his medical care and nursing facility.
That battle was settled in March when Charlene Conway was granted conservatorship over her ailing husband.
Kelly Conway is one Conway's children from his first marriage, to Mary Anne Dalton, which ended in divorce in 1978. He married Charlene Conway in 1984.
Kelly Conway had gone to court to prevent Charlene Conway from transferring him from an "excellent skilled nursing facility" to one she worried provided fewer services. She also complained she was prevented from seeing her father.
But Kelly Conway said she was "very happy" with the outcome of the case. She said she was allowed to see him and to meet with his caregivers, according to Neal's statement.
Conway is best known for his comic turns in skits, often in partnership with comic actor Harvey Korman, on Burnett's sketch comedy/variety show in the 1970s. The duo spent the rest of their careers often performing together until Korman's death in 2008.
But they are best remembered for their 10 years on "Burnett," when Korman hilariously struggled to contain his laughter at Conway's antics, even in the middle of their skits.
Conway's acting credits span six decades. He made a comic splash as Ensign Charles Parker on "McHale's Navy," the 1960s series about the wacky crew of a PT boat during World War II. After that, he was a regular guest star on dozens of TV comedy series and sitcoms until as recently as 2015.
He also voiced the role of Barnacle Boy for the hugely popular children's series "SpongeBob SquarePants."
His work on "Burnett" yielded him four Emmys and a Golden Globe. He won two other Emmys for more recent guest roles on "Coach" and "30 Rock."
Conway's 2013 memoir, "What's So Funny? My Hilarious Life" (with co-author Jane Scovell), looked back on 50 years in show business.
Conway, who largely retired in 2016, was unable to appear in the 50th-anniversary celebration of "The Carol Burnett Show," which aired in December 2017.
Born Thomas Conway in 1933 in the Cleveland suburb of Willoughby, he attended Bowling Green State University and served in the U.S. Army. He got his career start on local TV in Cleveland in the 1950s, performing comedy spots on a late-night movie show.
He was spotted by comedian/actress Rose Marie of "The Dick Van Dyke Show," who got him an audition for "The Steve Allen Show," where he became a regular in the early 1960s. Rose Marie died in 2017, but one of her children tweeted about Conway on the Rose Marie-Official Twitter account.
"So sad to hear about Tim Conway. 'Discovering' Tim and managing him for a time, was a source of tremendous pride for Mother. He was, after all, one of the funniest men on the planet! My heart goes out to his family."
So sad to hear about Tim Conway. "Discovering" Tim and managing him for a time, was a source of tremendous pride for Mother. He was, after all, one of the funniest men on the planet! My heart goes out to his family.https://t.co/1Q7NULVFpY— Rose Marie-Official (@RoseMarie4Real) May 14, 2019
Contributing: The Associated Press
This article originally appeared on USA TODAY: Tim Conway dies at 85; Carol Burnett is 'heartbroken'