Eastman, 83, was the brother of Linda Eastman McCartney, The Beatles star’s first wife, who died in 1998.
Sharing a photo of the pair doing yoga poses, McCartney wrote on Twitter: “My dear brother-in-law, John Eastman, has passed away. Having known each other for over 50 years it is an extremely sad time for me, and our families.”
The musician continued: “John was a great man. One of the nicest and smartest people I have had the good luck to have known in my life. Not only did he help me massively in my business dealings as my lawyer but as a friend he was hard to beat.
“His sense of humour always shone through in everything he did and his devotion to his family was supreme. We had so many fun times together through the years, but when the time came to be serious, he was unbeatable.
“There is so much more that can be said of his incredible qualities, but words can hardly describe his passion for life and our affection for this amazing man.
“He will be sorely missed but always held dear in the hearts of those of us who knew and loved him.”
McCartney concluded the post with a simple goodbye: “See ya Johnny! Love Paul.”
John and his father Lee served as McCartney’s lawyers, steering the star through The Beatles’ break-up.
Eastman is survived by his wife Jodie Eastman.