Haigh, who was set to become chief executive under Cellino, revealed that he had been given no choice but to leave the club following statements made by the Italian.
Cellino was critical of Haigh in an interview on Sunday in which he stated his intention to sack him once his purchase of a majority stake in the club was complete.
On Wednesday, a spokesman for Haigh said the 36-year-old still intended to take up the role of CEO as agreed as part of the contract between previous owners GFH Capital and Cellino's Eleonora Sport company.
However, in a statement released on Friday, Haigh said: "Owing to various statements made by and on behalf of the new majority owners of Leeds United FC, I am left with no alternative than to resign as managing director of the club.
"This is a matter of particular regret to me since I was the person who first introduced Eleonora Sport to the club's owners. I also gave them my full and constant support in the Football League's lengthy approval process.
"I am not yet, due to confidentiality obligations, in a position fully to respond to various statements which have been made about me over recent months. As soon as I am I will address the various issues - obviously a great deal has happened these past two years.
"As is well known by those in or close to the club, my unstinting support of Leeds United throughout my time at the club has extended to loaning the club money to ensure that tax, players and staff were paid and to underpin its continuing viability - loans which to this day remain in the club.
"I also searched endlessly for suitable investors, whilst at the same time not taking any payment that was due. In addition to all this I had to deal with and manage what can only be described as the crazy situation of very limited support from those who should have supported the club and the management, whilst at the same time having little or no decision making ability.
"On occasions this resulted in my paying club running expenses on my personal cards and last minute dashes to wire personal money to the club to pay the HMRC.
"Notwithstanding, I did everything which was in my very limited power to take the club forward, to engage with the fans and the community and to keep the promises which were made to its great fans.
"I very much hope that the new owners will do the same and will deliver on the many promises and contractual commitments they have made to Leeds United, the players, the manager, the staff and fans and will run the club in the manner of the fit and proper owners they have been judged to be.
"I have arranged for Sport Capital to convert some of the loans made to the club into shares and once complete, those shares will be given to the fans, so it truly will be the fans' club.
"Leeds is a fantastic club in a fantastic city and I wish Elenora Sports, (manager) Brian McDermott, the players, staff and fans all the very best for the future."
Meanwhile, West Yorkshire Police said it was investigating allegations of
theft at the Sky Bet Championship club.
It is understood the allegations centre on the installation of surveillance
equipment in the stadium's boardroom and toilets.
Detective Superintendent Pat Twiggs, head of crime for Leeds, said: "We can
confirm that police are investigating an allegation of theft relating to Leeds
United Football Club following a report made by the club made on April 9.
"The investigation relates to an allegation of club funds being
misappropriated for the purchase of technical equipment.
"Inquiries are at an early stage and we are not in a position to give any
further information about the nature of the allegation."