There were rumblings that the businessman would be forced to step down from his position after the Football League cleared the way for Cellino to become director this week.
The Italian's company Eleonora Sport Ltd was initially blocked by the league after Cellino was convicted of tax evasion in Sardinia.
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 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.
"Leeds is a fantastic club in a fantastic city and I wish Elenora Sports, Brian McDermott, the players, staff and fans all the very best for the future."
Haigh, whose Sport Capital consortium failed to take charge of the club in January, has been on the Leeds board since December 2012.