Haigh, who had been set to become chief executive following Cellino's takeover, had been on the board since GFH took over at Elland Road in December 2012.
He attempted to take charge of the club through his Sport Capital consortium but the deal collapsed in January.
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
Leeds are currently 16th in the table after five straight defeats.
Haigh said in a statement: "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 (Cellino's company) 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."
His departure comes on the day that West Yorkshire Police confirmed they were "investigating allegations of funds being misappropriated for the purchase of technical equipment".
There is no suggestion that the two events are linked.
Head of crime for Leeds, Det Supt Pat Twiggs said the investigation, which is believed to involve payment for unauthorised hidden cameras, had started after a report was made by the Championship club on Wednesday.
He added: "Enquiries are at an early stage and we are not in a position to give any further information about the nature of the allegation."

The incident at Neil McEvoy's office took place on Thursday morning. The AM said no valuables were taken but files had been "gone through".
He said metal bars had been cut to access the property at Cowbridge Road East.
South Wales Police said an investigation had been launched.
"For a long period, there has been a concerted campaign of political harassment against me," the South Wales Central AM claimed.
He said that less than 12 months ago his home had been "ransacked", with paperwork taken.
Mr McEvoy said that in the early hours of Thursday "the criminals involved went through two outside doors, down a lane, removed a screwed in hard wood cover on the back window, cut through a PVC window frame and cut through metal bars to get in the window".
"Once again, valuables were ignored in the office, but drawers were broken into and files gone through in the back office which wasn't alarmed.
"Thankfully, they couldn't gain access to the main part of the office without tripping the alarm."
"I put in place new safety procedures for staff at the end of last term, because I am aware of serious cages being rattled," he said.
"Some people would like to silence us, but this intimidation has just added to our motivation."
A South Wales Police spokesman said that the break-in was reported after 09:00 GMT on Thursday.
He said that investigations were ongoing and that no arrests had been made.
A Plaid Cymru spokeswoman said: "Neil McEvoy has now been the subject of two break-ins, one at home and one at his office.
"Plaid Cymru is supporting him and his staff at this time and is keen that the police establish whether these intrusions are linked.
"All elected representatives and their staff deserve to operate in a democratic way without fear of harassment or intimidation.
"The National Assembly is working with Neil and his staff to ensure their safety and security, and we strongly urge that this continues."â€Ž
Leeds United managing director David Haigh has resigned from the Championship club's board the day after Massimo Cellino completed his takeover.
[NEXT_CONCEPT]
A Plaid Cymru AM has claimed he has been a victim of "political harassment" after a break-in at his constituency office in Cardiff.