The 17-year-old ball-boy, whose name has been revealed as Charlie Morgan, was attempting to smother the ball to waste time as the clock approached the 80th minute.
Hazard lost patience and appeared to kick the ball-boy in the ribs. The Belgian international was subsequently sent off.
The misdemeanour will not result in a police charge but could lead to a severe sanction from the English Football Association.
The moment quickly went viral online, as both #hazard and #ballboy became worldwide Twitter trends.
Morgan, who is son of the Swansea City director Martin Morgan, has worked as a ballboy at Swansea for six years, and tweeted earlier on Wednesday: “The king of all ball boys is back making his final appearance #needed #for #timewasting”.
The tweet would suggest Morgan’s time-wasting was a preordained tactic – although who sanctioned his behaviour, if anyone, remains to be seen.
The tweet currently has a massive 33,419 reweets and 5,265 favourites, while Morgan now has 75,650 followers – a number growing by the minute.
Hazard, for his part, stated after the match that he and Morgan had exchanged apologies in the Chelsea dressing-room: “The boy put his whole body on to the ball and I was just trying to kick the ball and I think I kicked the ball and not the boy. I apologise. The ballboy came in the changing room and we had a quick chat and I apologised and the boy apologised as well, and it is over. Sorry.”
The match finished 0-0, with many pundits citing Hazard's red card as the moment that sealed Chelsea's fate.
The Blues lose 2-0 on aggregate, and Swansea go through to their first major cup final. They will meet Bradford City in the final.