"Steve's extraordinary vision and leadership saved Apple and guided it to its position as the world's most innovative and valuable technology company," board member Art Levinson said in a statement.
No reason was given for Job's resignation, but his health problems, including a lengthy medical leave for a liver transplant in 2009 and his increasingly gaunt appearances at public events, fueled speculation he would have to give up the everyday running of the company he co-founded in 1976.
Cook ran Apple when Jobs went on medical leave and has essentially been running day-to-day operations since early this year with the company racking up record revenue and profit.
Jobs is seen as the heart and soul of Apple, with analysts and investors repeatedly expressing concern over how the Cupertino, California-based company would handle his departure.
"The board has complete confidence that Tim is the right person to be our next CEO," Levinson said.
"Tim's 13 years of service to Apple have been marked by outstanding performance, and he has demonstrated remarkable talent and sound judgment in everything he does," Levinson continued.
Jobs submitted his resignation on Wednesday and urged the board to implement its succession plan and name Cook as his replacement, according to Apple.