The Prince, who is president of the English Football Association (FA), said FIFA must show it "can represent the interests of fair play and put the sport first."
His strongly worded remarks, in a speech to guests at the FA Cup final, follow Sepp Blatter's re-election as FIFA president at its annual congress in Zurich on Friday.
Blatter survived a challenge from Jordan's Prince Ali bin al-Hussein
despite the arrests of seven leading FIFA officials in the Swiss city on Wednesday, and the announcement by U.S. authorities that 14 people were being indicted on allegations of taking $150 million in kickbacks dating back more than 20 years.
Blatter has not been accused of any wrongdoing as a result of the U.S. investigation, but his leadership of FIFA has come under severe criticism, led by European football's governing body UEFA, of which the FA is a key member.
Learn the lessons
Prince William said FIFA must learn the lessons of the corruption scandal which hit the International Olympic Committee (IOC) over the decision to award the 2002 Winter Games to Salt Lake City, when it emerged that IOC members had been offered inducements to support the city's bid.
"The events in Zurich this week represent FIFA's Salt Lake City moment, when the International Olympic Committee went through a similar period of serious allegations.
"FIFA, like the IOC, must now show that it can represent the interests of fair play and put the sport first," he said.
He called on key sponsors, as well as bodies such as UEFA, to put pressure on FIFA to make the necessary changes.
"Those backing FIFA, such as sponsors and the regional confederations, must do their bit to press these reforms -- we are doing football and its fans no favors if we do not.
"I have no doubt that when FIFA reforms, its mission to spread the benefits of the game to more people, especially those in developing countries, can only be enhanced."
World Cup bidding probe
Aside from the arrests prompted by the U.S. investigation, Swiss authorities have also launched a separate probe into the bidding process around the award of the 2018 and 2022 World Cups to Russia and Qatar.
Prince William was a prominent backer of England's failed bid to host the 2018 competition, along with British Prime Minister David Cameron and soccer icon David Beckham.
Former Manchester United chief executive David Gill was also a key member of the 2018 bid team; he has refused to take up his place on FIFA's executive committee in protest at Blatter's re-election, saying it would be "futile" to serve under the 79-year-old.
Prince William said he fully backed Gill's decision: "I know I join with all of you in commending David Gill for his decision to stand down from the ExCo, and to lead by example by doing so," he told the audience.
UEFA emergency meeting
UEFA is to meet in emergency session next Saturday ahead of the Champions League final in Berlin to consider its next move.
FA chairman Greg Dyke, who attended the FA Cup final with Prince William, told the BBC Sunday that a boycott of the 2018 World Cup would be considered, but that it would be futile for England to act alone.
"There's no point boycotting on our own, but if the rest of Europe decided to boycott we would join them," he said.
A defiant Blatter told Swiss television Saturday that it was "no coincidence" that arrests were made ahead of the FIFA Congress in tandem with calls by UEFA chiefs for him to stand down, suggesting the move was an attempt to derail his presidential bid.