Russia then earned an absolute majority of 13 votes, with Spain-Portugal on seven and Netherlands-Belgium picking up two.
"I'm very disappointed, as are the rest of the team," said Prince William, who spent much of the week lobbying for support and delivered an impressive speech during England's presentation.
"The guys are quite down at the moment but they put their whole backs into it. I'm so glad they gave it as much effort as they can.
"We gave everything that we could. I'm sorry for the fans back home that this happened."
Prime Minister Cameron described the result as "bitterly disappointing" and felt there was not much more England could have done to convince Fifa that theirs was the best bid.
"I think, according to Fifa, we had the best technical bid, the best commercial bid," stated Cameron. "No-one could identify any risks coming to England. But it turns out that's not enough.
"A huge credit to the team - they worked extremely hard. I would pay huge tribute and credit to David Beckham, who gave an inspirational speech, and Prince William, who worked incredibly hard.
First round: England 2 votes (eliminated), Netherlands-Belgium 4; Spain-Portugal 7; Russia 9 (no absolute majority)
Second round: Netherlands-Belgium 2; Spain-Portugal 7; Russia 13 (Russia obtain absolute majority)
"It's hard to see what more you can do. It turns out having the best technical bid, the best commercial bid and having a passion for football isn't enough. I was hoping to see a World Cup in England in my lifetime."
The next World Cup that England is likely to be entitled to bid for is 2030. But Fifa's apparent determination to open the game up to new markets suggests the wait may go on even longer.
Beckham, who has 115 England caps, told BBC Radio 5 live: "I'm obviously disappointed. Our bid team have done everything possible. We couldn't have got a better bid together.
"We wish Qatar and Russia the very best. They're two great countries and I'm sure they'll make Fifa very proud.
"It's hard not to come away with the World Cup in 2018. Lots of congratulations to the team. A lot of hard work has been done - the Prime Minister's hard work, Prince William's hard work.
"Apologies that we couldn't bring the World Cup to our country because there's no better and more passionate fans in the world than in England.
"It's just disappointing that we can't bring the World Cup to them in 2018."
Former England captain and BBC pundit Alan Shearer felt England had the strongest bid on the table and questioned whether his country would ever be in a better position to host football's showpiece competition.
Prime Minister Cameron spent much of the week campaigning for votes
"I was hoping that I might see a World Cup in my lifetime in England," said the 40-year-old former Newcastle striker. "Everyone was very happy and very confident with the bid we put in. We couldn't have done any more.
"If we haven't got it this time, when are we ever going to get it? I felt very confident with Prince William and the Prime Minister behind us."
Though England is blessed with a series of excellent stadia, good transport links and policing, it is thought a number of behind-the-scenes factors possibly went against them.
And the BBC's recent Panorama investigation - broadcast on Monday - accused three Fifa executive committee members of accepting "corrupt" payments and alleged that Fifa vice-president Jack Warner attempted to supply ticket touts.
Michel Platini, president of European football's governing body Uefa, insisted the documentary would not alter members' votes or wreck England's chances.
But he did claim that the British media's arduous relationship with football's international governing body could jeopardise England's chances of success.
Hayatou, Brazil's Ricardo Terra Teixeira and Nicolas Leoz of Paraguay were all accused by Panorama of taking payments.
Mr Warner was the subject of accusations that he "ordered [2010 World Cup] tickets costing $84,240 [£53,962] from the Fifa ticket office but the deal subsequently fell through".
The four men were all part of the electorate who voted.
Current England captain
giving his reaction on Twitter, said: "The timing of the Panorama programme was bad taste, fact."
Prince William 'extremely disappointed' as England miss out
Beckham added: "I've heard the rumours that we lost due to the British press. I hope that isn't the reason. I believe in a free press and they are incredibly supportive of the game I love."
Ex-England striker and BBC Match of the Day presenter Gary Lineker said: "Fifa have always said they want to push the frontiers - I was surprised they went with two new countries but there were a lot of good strong bids.
"English football will carry on regardless. We're very strong in our country in terms of our league. It could have been a huge boost. It's a real shame we'll never get a chance to prove that."
London Mayor Boris Johnson was disappointed the city did not have more to celebrate after winning the bid to host the 2012 Olympics.
"This is a tremendously disappointing result," Johnson reflected. "We put together a cracking bid, our technical specification was top notch and our stadiums would have been packed to the rafters. Londoners love football."
England bid board member Lord Sebastian Coe admitted Russia's presentation on Thursday had been shrewd and may have helped seal their success.
The Russians pointed out the World Cup had yet to be staged in eastern Europe, compared with the 10 times it has been held in the west of the continent.
Serious questions have to be answered as to how we can learn lessons from these significant setbacks to build a better future
Shadow Secretary of State for Sport Ivan Lewis
"It was a very similar concept that Rio went for in Copenhagen [when they won their bid for the 2016 Olympics] just a few months ago," Lord Coe told 5 live. "I think that worked quite strongly for them today."
Former Arsenal vice-chairman David Dein was also integral to England's bid team, and was angered that "the best team didn't win".
"It was always going to be difficult and it wasn't made any easier by certain elements. I'm not going to go into that now," he said.
"When the technical study and the economic reports came out, we were one of best - if not the best - and I think you'll agree that the presentation today was top class, so that makes it all the more disappointing that we didn't do better."
Culture Secretary Jeremy Hunt told BBC Sport he was "absolutely devastated. We would have delivered simply the best World Cup for 2018 but congratulations to Russia. It's absolutely gutting. A very sad day, a heart-wrenching day for our country."
In a statement from shadow Secretary of State for Sport Ivan Lewis, the Labour Party called for an independent root and branch inquiry into the failed bid.
"Despite the impressive efforts of Team England, supporters will be bitterly disappointed at England's failure to land the 2018 World Cup," said Lewis. "This follows on from the England team's poor performance in South Africa.
"Serious questions have to be answered as to how we can learn lessons from these significant setbacks to build a better future.
"The Coalition and the football authorities should now set up an independent root and branch inquiry into all aspects of how our national game is run."
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