“We’ve been on the air for almost 23 years, and I’ve had to do some hard monologues along the way,” he said. “This one’s the hardest.”

Kimmel did not cite a cause of death but thanked the staff at UCLA Medical Center and Sherman Oaks Hospital in Los Angeles for caring for Mr. Escobedo after an “awful few months.”

Mr. Escobedo had been the bandleader of Cleto and the Cletones, the house band on “Jimmy Kimmel Live!” since the show debuted in 2003.

Kimmel described an inseparable youth together in Las Vegas, where Kimmel’s family moved in 1977 from New York. He was aged 9 when he befriended Mr. Escobedo and said they had been close throughout their lives.

Marking Mr. Escobedo’s 50th birthday on the show in 2016, Kimmel said the bandleader had inflicted the kind of “torture” on him during their childhood that only an older brother could. Mr. Escobedo once glued the steering column of Kimmel’s homemade go-kart as a prank, and sent the young Kimmel careening into garbage cans while riding in a sidecar welded to the frame of his bicycle.

Kimmel recalled Tuesday how Mr. Escobedo would “moon” passersby from the back of the Kimmel family car, pressing his backside against the rear window above a bumper sticker that read: “the family that prays together, stays together.”

“It’s funny because as an adult, he was not wild,” Kimmel said. “He was a dad who, if we did go out, we’d go on a fishing trip. He liked to stay home with his family. He never missed a day of work.”

Mr. Escobedo, who Kimmel described as “a phenomenal saxophone player from a very young age,” became a professional musician, playing with Paula Abdul and Marc Anthony and releasing an album, “Cleto,” in 1995. The album was not a commercial success, and he went back on the road touring with a string of well-known artists.

“Cleto was never really a pop star,” Kimmel said. “He was more of a serious musician. He was a great saxophone player. He loved jazz and R&B and different kinds of music than what the record company wanted from him.”

Kimmel pushed for Mr. Escobedo to be appointed bandleader when he found out he would be hosting his own talk show. Mr. Escobedo’s father, Cleto Escobedo Jr., is also a saxophonist in the band and played during Tuesday’s show.

According to Kimmel, Mr. Escobedo’s father gave up a promising career with a Texas band, Los Blues, because he didn’t want to go on tour and be away from his son. After Mr. Escobedo was born in Las Vegas on Aug. 23, 1966, his father quit and worked as a bus boy at Caesars Palace until one day Sammy Davis Jr. recognized him from the band and got him a job as his room service butler, Kimmel recalled.

Kimmel also gave Mr. Escobedo’s mother a shout-out in the audience on Tuesday night. “Sylvia comes to work here every day, sits in that seat — despite the fact that she does not work here. She does not have a job here. She comes and stays all day to be here with her husband and son. She’s working those rosary beads in the audience every night.”

Mr. Escobedo is survived by his parents, his wife Lori and their two children, Cruz and Jesse.

“The fact that we got to work together every day is a dream neither of us could ever have imagined would come true,” Kimmel said in his Instagram post.

On his show Tuesday night, he said, “There were thousands of houses for sale in Vegas; my parents happened to buy one that was right across the street from this kid that I would just fit together with so perfectly.”