Hazard, who was sent off, later met the 17-year-old named locally as Charlie Morgan, son of the Swansea City director Martin Morgan, in the Chelsea dressing room and apologised for the incident.
Swansea, who won 2-0 at Stamford Bridge in the first leg, held on for a goalless draw at Liberty Stadium to set up a final against Bradford City but the main talking point was the bizarre altercation that led to the Chelsea midfielder's red card 12 minutes from time.
Belgian Hazard said he had not intended to boot the ball boy.
"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," Hazard told Chelsea TV, according to the club's official website.
"I apologise. The ball boy 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 red card shown by referee Chris Foy will lead the 22-year-old to be banned for three games, though there may be scope for that sanction to be extended.
"They have apologised to each other," Chelsea's interim manager Rafael Benitez said. "They knew they were both wrong. He (the ball boy) was wasting time. Hazard wanted to get the ball back quickly and he was kicking. They both made mistakes.
"We will deal with this internally. He was frustrated and just wanted to get the ball back, but we will analyse it and see what happens."
Premier League players reacted to the incident on social media with Manchester United defender Rio Ferdinand condemning the Belgian's actions, while Stoke striker Michael Owen felt both parties were to blame.
"I'm not the authorities but in my eyes it's a red for all those asking. Us players do put refs in bad spots at times unfortunately!" Ferdinand tweeted.
"Both the kid and Hazard were in the wrong. Not having Hazard tried to hurt him though. He just tried to toe poke it out of his grip," former England forward Owen wrote on his Twitter feed.
Both clubs confirmed that Hazard will not face criminal charges for his part in the astonishing incident.
The ball boy was quickly ushered away from the tunnel area after the final whistle having been interviewed by police officers.
A Swansea spokesman confirmed no action would be taken, and that Chelsea had invited the 17-year-old into their dressing room.
"The police have interviewed the boy and his father and there are no charges going to be pressed, as far as they are concerned that is the matter closed," he said.
"The ball boy has been into the Chelsea dressing room after the game and shaken hands with Eden Hazard. The likes of John Terry and Frank Lampard were exemplary in their behaviour in welcoming him into the dressing room.
"Chelsea asked if he could come in and our kit lady took him in and they shook hands, he is fine.
"As far as we are concerned the matter is closed. He has been a ball boy here for six years with no incidents here at the stadium."