Cook’s decision to step away from the chief executive role follows months of speculation that Apple was looking for a successor.
He described the job as “the greatest privilege of my life” and during his tenure he led the company to become one of the most valuable in the world.
Cook described Ternus as a “visionary” executive with “the mind of an engineer, the soul of an innovator and the heart to lead with integrity and honour”.
“He is without question the right person to lead Apple into the future,” Cook added.
Ternus emerged as a favourite to replace Cook last year, after another long-time executive, chief operating officer Jeff Williams, left the company.
During his quarter century at Apple, Ternus has worked on essentially every major product the company has released, including every generation of the iPad, many generations of the iPhone, and the launch of AirPods and the Apple Watch.
He also oversaw the transition of Mac computer processors to Apple’s own silicon.
Ternus also worked under Jobs, before his retirement in 2011 and death six weeks later. In a statement on Monday, he referred to Cook as his “mentor.”
“I am filled with optimism about what we can achieve in the years to come,” Ternus said.