"Remembering my friend Steve on his birthday," Cook wrote in the first of a pair of tweets. "'Stay hungry, Stay foolish.' We honor him by continuing the work he loved so much."
In a second post to his account, Cook shared another quote from Jobs: "Details matter. It's worth waiting to get it right."
Jobs was born on Feb. 24, 1955, in San Francisco, Calif. He passed away in October of 2011 at the age of 56 after a long fight with cancer.
Only a few months before, Jobs permanently resigned from his position as CEO, handing over the reins of the company to Cook that August. Prior to that, Cook served as Chief Operating Officer under Jobs.
Last week, the U.S. Postal Service announced it will commemorate Jobs with his collectible postage stamp next year. A movie about Jobs' life, penned by Academy Award winning screenwriter Aaron Sorkin, is also in the works.