Hazard was sent off for kicking 17-year-old Charlie Morgan during the Blues' 0-0 draw at the Liberty Stadium last Wednesday.
Morgan, in retrieving the ball with Swansea in a winning position, lay down on it as Hazard seeked to get hold of it himself - so the Chelsea star tried to kick it out from under him.
It had been suggested that the FA were considering handing down a lengthy suspension to the Belgian, but the governing body ruled that the usual three-game ban for violent conduct should stand instead.
The Blues had argued that Hazard moved quickly to apologise for the fracas and should therefore be treated leniently.
They 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.
Swansea reached the final and will face Bradford City at Wembley on February 24.