Payne, who shot to global fame with boy band One Direction, died at age 31 after falling from a third-floor hotel room balcony in Buenos Aires on Wednesday, triggering tributes from fans around the world.
A joint statement was signed by singers Harry Styles, Louis Tomlinson, Zayn Malik and Niall Horan.
"We're completely devastated by the news of Liam's passing. In time, and when everyone is able to, there will be more to say. But for now, we will take some time to grieve and process the loss of our brother, who we loved dearly," Payne's former bandmates said.
They also said their thoughts were with the English singer's family, friends and fans, concluding: "We will miss him terribly. We love you Liam."
In a separate statement posted on Instagram on Thursday, Styles said Payne's greatest joy was making people happy.
"The years we spent together will forever remain among the most cherished years of my life," Styles said.
Malik also posted on Instagram about the late singer, along with a photo of them dozing off next to each other.
"I lost a brother when you left us and can't explain to you what I'd give just to give you a hug one last time," he wrote.
View this post on Instagram A post shared by Zayn Malik (@zayn)
Along with a picture of Payne embracing him at a gig, Tomlinson wrote he was beyond lucky to have had Payne in his life.
"I'm really struggling with the idea of saying goodbye," he said.
"I would have loved to share the stage with you again but it wasn't to be."
Music mogul Simon Cowell, who signed One Direction to his label, posted his tribute on Instagram on Friday, saying he was "truly devastated" and "heartbroken".
"And I want you to know how much love and respect I have for you. Every tear I have shed is a memory of you," he said.
Other celebrities have paid tribute to Payne, with former footballer David Beckham describing him as a generous person and musician Ed Sheeran saying he was at a loss for words and urging everyone to be kind.
Prosecutors in Buenos Aires said on Thursday an initial search suggested Payne's fall from the hotel balcony came after substance abuse.
Police earlier said they had found the room in "total disorder" in a search after his death, with various things smashed and medications scattered around.
Sharon Osbourne, former judge of TV talent show X Factor, where One Direction was formed, said on Instagram that the music industry had let Payne down.
"Where was this industry when you needed them? You were just a kid when you entered one of the toughest industries in the world. Who was in your corner?" Osbourne asked.
Auditions for Britain's Got Talent in Blackpool, England, were postponed on Thursday "due to the tragic passing of Liam Payne," Applause Store, the ticket provider for the show, said in an Instagram story.
Cowell is the creator of both X Factor and Britain's Got Talent.
