Cook, 62, suffered a major heart attack last season but was back at work before the end of the campaign as Durham celebrated their third county title, all won under his regime.
He will retain his current title and head the coaching department but will now switch his focus to the team's youth sides, academy and second XI.
Lewis, who stepped in as interim coach during Cook's recovery, will work with the senior side, with Alan Walker assisting as bowling coach.
Cook said: "The club's achievements over the last 21 years are nothing short of remarkable and not many would have anticipated three County Championship titles in such a short time.
"The fact that this success was based on the development of North East cricketers makes it all the more special.
"There is a lot of young cricket talent in the Durham youth squads and we hope that many of these players come through to represent to first team and hopefully bring glory to Durham in years to come."
Lewis added: "I am delighted to be given the opportunity to continue this role and I am excited for the challenge ahead.
"This is a very interesting time for the club with the evolution of the side and the increased opportunities for home-grown players to shine.
"Having been second XI coach for many years I have worked with a lot of the young players coming through and now beginning to perform in the first team.
"I will hopefully develop the partnership with Paul (Collingwood), we work extremely well on and off the field and hopefully we can bring success to the club in 2014."
Cook's success with the county were marked by chief executive David Harker.
He said: "On behalf of the club I would like to acknowledge Geoff's incredible achievements with the first team over the past six years.
"Durham County Cricket Club has delivered outstanding results under Geoff's leadership and we are thrilled that Geoff will now focus his considerable expertise on our youth development programme that is so important to the continued success of the club."