More recently, the Scottish-born actor was known for his long-running role as a medical examiner on hit TV show NCIS.
"We will miss his warmth and endearing sense of humor that lit up any room or soundstage he stepped onto," the NCIS account said on social media.
The actor also starred in Colditz, The Invisible Man and Sapphire & Steel.
"David was a gifted actor and author, and beloved by many around the world," the NCIS tribute said.
"He led an incredible life, and his legacy will forever live on through his family and the countless hours on film and television that will never go away."
Obituary: Actor who got more fan mail than Elvis
His role as mysterious Soviet agent Illya Kuryakin in The Man from U.N.C.L.E. won him many fans and made him one of the decade's biggest TV stars.
The series ended in 1968, but not before he received several Emmy and Golden Globe nominations.
Another NCIS actor, Wilmer Valderama, shared a still of himself acting opposite McCallum, noting it had been an "immense honour" to share the screen with him. "Your professionalism and ability to effortlessly take every one of us on a journey through your art will be felt forever," he wrote.
Emily Wickersham added on her Instagram story: "Had the pleasure of working with David for many years and he was the utmost professional and a true legend."
McCallum's role on NCIS came after he appeared for a role in the show JAG, which led to the NCIS spin-off. NCIS itself later went on to generate other NCIS shows, including NCIS: Los Angeles and NCIS: New Orleans.
McCallum also found work as a voice actor for children's cartoons and video games.
'No one did it better'
The actor died in New York on Monday. His death was said to have been due to natural causes.
NCIS co-star Michael Weatherly was among the others paying tribute.
Sharing a signed photo of McCallum alongside Steve McQueen in The Great Escape, Weatherly wrote: "David McCallum made every moment count, in life and on set. Let's raise a jug and celebrate a funny fantastic authentic man.
"I've only got three autographs. [Sean] Connery, Tony Bennett and McCallum," he continued. "I felt the same way as Steve McQueen in this picture from The Great Escape: Wow! It's David McCallum! No one did it better."
Dr Dre sample
In the 1960s, he recorded four albums for Capitol Records alongside producer David Axelrod.
One of his tracks, The Edge, was sampled to memorable effect by Dr Dre on the US rapper's 2000 hit single The Next Episode, featuring Snoop Dogg. McCallum's instrumental original also went on to appear on the soundtracks to 2008 video game Grand Theft Auto IV and 2017 film Baby Driver.
In a statement released by CBS, his son Peter McCallum said: "He was the kindest, coolest, most patient and loving father. He always put family before self.
"He was a true renaissance man - he was fascinated by science and culture and would turn those passions into knowledge.
"For example, he was capable of conducting a symphony orchestra and (if needed) could actually perform an autopsy, based on his decades-long studies for his role on NCIS."
Four years later, he played an RAF officer in the BBC's Colditz, set in the German World War Two prisoner of war camp.
In The Invisible Man, he starred as Dr Daniel Westin, a scientist who inadvertently discovered the secret of invisibility.
And at the end of the 70s, he teamed up with Joanna Lumley as a time-travelling agent, assigned by an unknown authority to safeguard time itself, in Sapphire & Steel.
On the big screen, McCallum had roles in films including The Great Escape, The Greatest Story Ever Told and A Night to Remember.
He also guest starred on TV series Perry Mason and The Outer Limits.
Born in Glasgow to parents who were classical musicians, McCallum initially pursued a career in music before finding work as an actor.
