Roy Dobson, from Chorley in Lancashire, had linked the account to his credit card, and son Alfie was able to spend £99 on more than 50 in-app purchases.
He said he wants to warn other parents how easy it is to run up huge bills as the charges "racked up and racked up".
Technology giant Apple has agreed to refund the family.
Mr Dobson said: "The first time he spent £700 in less than five minutes, then £1,100 in half an hour and it just racked up and racked up, all on the same game."
His wife Jill said Alfie is only allowed to play on the iPad at weekends.
She added: "He's bought things in the past for 99p or £1.49, but he's always asked and then he saw this at £99 and was just curious as to what you would get for £99.
"It was just to get better in the game, there's nothing to show for it, I didn't even know you could buy things for £99.
"It's scary. He said the game was that good he couldn't stop, but he only thought he pressed it a few times."
Games journalist Guy Cocker said: "These games were initially released for £8.99 or £9.99 but the games companies found they were able to make more money from them by offering in-app purchases, so the idea is that you download the initial game for free and then you pay for items or extras as you play.
"They are called micro-transactions as they usually cost a matter of pence.
"But it's completely feasible that someone could spend a lot of money unlocking items."
Apple did not comment on this specific case but did say that its parental controls and "Ask to Buy" features are a good way for parents to protect young users from making unauthorised purchases.
An 11-year-old boy unwittingly spent almost £6,000 playing online games over two weekends after accessing the family's iTunes account.