The Football League recently decided that Cellino had failed the Owners' and Directors' Test and barred him from taking over the club on the grounds of his conviction in a court in Italy.
But an independent QC has overturned the decision and Cellino can now take over the club.
Cellino said: ""Justice has been done. I'm happy for the Leeds fans. A lot of hard work has gone into this. We need to start winning games!"
In a statement the League said it is "disappointed" by the decision and is considering whether to take the matter further.
A Football League spokesman said: "We are disappointed at the outcome of the appeal hearing, however we would like to thank the independent QC for his diligence in reviewing this decision.
"This was never about individual personalities, but instead was a matter in which we were obliged to uphold the integrity of our regulations having considered the issue in detail with our advisors.
"It was always an extremely complex matter in which a different interpretation of a judgment made under Italian law could lead to an entirely different outcome in the context of our regulations. Ultimately this has proven to be the case.
"The independent QC has concluded that Mr. Cellino's recent conviction in the Sardinian Court did not involve conduct that would 'reasonably be considered to be dishonest' based on information available to him at the current time.
"The Football League will now consider the findings of the hearing."
Cellino's Eleonora Sport will take a 75% stakeholding in the club while former majority shareholders GFH Capital and investors will retain 25%.
Cellino will become the club's President, heading a new-look board at Elland Road.
Leeds manager Brian McDermott, who Cellino attempted to sack at the end of January, welcomed the appeal victory.
"I would recommened the stability of a man who has got clout," said McDermott. "I've spoken to him on a number of occasions and, for me, the most important thing is not the position of the manager, or any individual player, it's Leeds United Football Club and that going forward.
"I can work with Massimo, we'll see just wait and see what the outcome is. It's a prerogative of any owner, in any situation, to choose the manager he wants to work with. I absolutely respect that."
Managing director David Haigh said: "This is a significant day in the history of Leeds United. Since GFH Capital took over the club in December 2012, the stated aim has always been to introduce strategic investors to ensure a sustainable future for the club.
"I have met with many potential investors and worked closely with them, and I realised very quickly that Massimo is someone who has the attributes to take this club forward.
"He has a proven track record in Italy, and I believe the introduction of Eleonora Sport, coupled with Massimo's drive and determination, will be key to a successful and sustainable future for this great football club.
"I'd like to thank the Football League, GFH Capital, and of course, Massimo and Eleonora Sport for all their hard work over the past few weeks in ensuring that this deal was able to progress."
Mike Farnan, who heads up the Yorkshire-based Together Leeds consortium, said: "I'm delighted for Massimo."
It was reported earlier this week that Cellino and Together Leeds would join forces and Farnan added: "Massimo's lawyers have told us that we will continue our meeting next week. This is a joint opportunity.
"We can now look to work together to get the club moving forward again. This has been dragging on for too long.
"From the Football League standpoint, and I don't know the ins and outs of all the legalities, it creates a lot of questions."
"We'll have to wait and see what they decide to do.
"But let's get moving. I'm really pleased for the fans. I'm so pleased for the fans. It's an exciting package that we can put together."