With time running out, Hazard was sent off after kicking out at the prostrate youth who was lying on top of the ball at the side of the pitch at the Liberty Stadium.
South Wales police questioned the boy and his father after the game but will not take the matter any further.
Hazard told Chelsea TV: "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 apologize.
"The ballboy came in the changing room and we had a quick chat and I apologized and the boy apologized as well, and it is over. Sorry."
Chelsea boss Rafa Benitez added: "The ball boy knows he was wasting time. We are disappointed to lose the player but we cannot change things. They both recognize there was a mistake.
"The boy was apologizing for time wasting. Hazard was frustrated and tried to get the ball. He was kicking the ball and getting the ball.
"We can analyze it for half an hour but we know that both are wrong."
A Swansea spokesman said: "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.
"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 behavior 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."
It appears Hazard has escaped police action but the Chelsea midfielder must wait to see what action the Football Association may take against him.
The 21-year-old could yet face a lengthy ban, though it is unlikely to be comparable to the nine months Manchester United's Eric Cantona missed after the most high-profile incident of a similar vein in recent years, his infamous kung-fu kick at a Crystal Palace fan in 1995.
In a further twist to the controversy Charlie Morgan, the ball boy in question, had apparently posted on his Twitter account just hours before the game: "The king of all ball boys is back making his final appearance £needed £for £time wasting".
And he appeared to suggest he had only been asked to work at the game due to the heavy snow in south Wales preventing someone else getting there.
"Haven't been doing it this season been ask to come back to run them because the person can't come in cos of the snow £manager", his tweet read.
His Twitter account had swelled to almost 50,000 followers after the game.
Benitez said that Hazard's actions would be dealt with internally by the club but reacted angrily when he was accused of attempting to condone Hazard's actions.
"I do not know what you expect from me," he said. "Do you think we are not disappointed with the situation, that we do not regret what happened? Do you think they have not apologized? They have done it so what can we do now? Do you want to change things? We cannot.
"The player spoke to the boy and he said sorry because it was a mistake. We know it was a mistake but we talked with the player and the boy so what do you expect?
"The boy was with us in the dressing room and everybody was clear and we know there was a mistake."