Hazard was sent off for kicking 17-year-old Charlie Morgan as he attempted to retrieve the ball during the Blues' 0-0 draw at the Liberty Stadium last Wednesday.
It had been suggested that the FA were considering handing down a lengthy suspension to the Belgian, but English football's governing body have ruled that the usual three-game ban for violent conduct should stand instead.
"Following a hearing earlier today, an independent regulatory commission was of the opinion the existing three-match sanction for this offence was sufficient."
The Blues wrote to the FA on Wednesday arguing that Hazard did not merit a longer punishment than usual for violent conduct.
The basis of their argument was that Hazard moved quickly to apologise for the fracas. The club are also thought to have pointed out that Swindon's Matt Ritchie was only booked for grappling with Oxford United ball boy Aidan Hawtin in a similar incident last year.
Hazard's kick provoked outrage in some quarters, but ball boy Morgan, who is the son of club director Martin Morgan, was also criticised for preventing the player from retrieving the ball by laying on it.
As a result, the FA have urged clubs to take all necessary steps to make sure such an incident does not occur again.