A statement published on the club's official website on Friday night confirmed the parting of ways.
In a statement, Cellino said: "Brian is a great manager and a great guy. He has been unfortunate to work in such difficult circumstances.
"I did not fully understand the mess he had to work in, and the broken promises he had to deal with, until I have got involved trying to turn Leeds around.
"He has been a gentleman to deal with in our discussions and has been very understanding of my wish to implement a new structure. His main concern and priority at all times has been the welfare and protection of Leeds United.
"I wish him well for the future where I am sure he will continue to have more success and thank him for his efforts in being a stabilising and unifying figure behind the scenes in very difficult circumstances.
"His honest efforts to guide us to the safety of mid-table when faced with many difficulties is appreciated by us all. He will always be a friend of Leeds United."
In the same statement, McDermott said: "It has been my great privilege to manage this great club and I have enjoyed the challenge immensely.
"However it is clear that Massimo wishes to implement a new structure and feels he will work more successfully for Leeds with a coach rather than a manager.
"I respect that Massimo must be fully supported in putting together his vision for the club so he can bring us the success we all want. It is my belief that when truly United we are stronger.
"Massimo wants to bring a new energy to the club so that we can return to where we belong as a healthy football club.
"To my players for their efforts I say a big thank-you. To all the staff and especially my great assistant manager Nigel Gibbs and my great friend Patrick Dolan I am especially grateful for their support.
"Finally to the incredible force that is the Leeds United supporters, I offer my heartfelt thanks.
"Your support of me and the players was always an inspiration and I urge everyone to get behind the team and the new owner next season to get us back to where we need to be, fighting for trophies and competing at the top-table of English and European football.
"It won't be easy, but Marching On Together we can get there.
"I wish Massimo and all my friends at Elland Road every success in the future."
McDermott leaves Leeds after guiding them to 15th place in the Championship in the 2013-14 season.
The former Reading boss replaced Neil Warnock as manager in April 2013 on a three-year deal.
The 53-year-old appeared to have been sacked in January before Leeds performed an astonishing U-turn a day later by announcing that McDermott had not been sacked and remained their first-team manager.
He has now left the club, having taken charge of 54 matches.
- Sports & Recreation
- Massimo Cellino
- Leeds United