The incident took place in the 78th minute at the Liberty Stadium as the Belgian tried to retrieve the ball after the youngster had lay on top of it behind the Swansea goal.
Hazard was then sent off by referee Chris Foy as Chelsea lost the tie 2-0 on aggregate.
"The boy put his whole body onto the ball and I was just trying to kick the ball and I think I kicked the ball and not the boy. I apologise," Hazard told the club's official website.
"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."
An investigation into the incident was swiftly dropped by South Wales Police on Wednesday night.