Hazard and the ball boy apologized to each other after the game, Swansea said in a statement.
"South Wales Police interviewed the 17-year-old ball boy with his father after the game and neither wished to press any charges," Swansea said. "It was an unfortunate incident which both parties regret being involved in."
With Chelsea needing two goals at Swansea's Liberty Stadium to force overtime in the two-game series, the ball bounced off an advertising board behind the goal line in the 79th minute.
The ball boy, wearing a Swansea sweatsuit, slowly walked over to pick it up.
Hazard ran up to reclaim the ball and the teen fell over, stomach toward the ground and the ball under him. The Belgian winger leaned over the teenager, tried to pull the ball free with both hands and then kicked at it under the youth.
After Hazard got the ball free, he picked it up and jogged back onto the field. The ball boy, who has worked for Swansea for six years, rolled onto his back and clutched the right side of his ribs as he was attended to.
Hazard was shown a straight red card by referee Chris Foy.
"Chelsea invited the ball boy into the dressing room after the game to shake hands with Hazard and both parties apologized," Swansea said. "The ball boy was very impressed with the welcome he received from the Chelsea players and as far as both clubs and individuals are concerned the matter is closed."
Hazard was quoted by Chelsea's website as telling Chelsea TV "the boy put his whole body onto the ball and I was just trying to kick the ball. I apologize. The ball boy 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."
The match finished 0-0 and Swansea advanced 2-0 on aggregate, setting up a final against fourth-tier Bradford at Wembley Stadium on Feb. 24. It will be the first major English final in the Swans' 100-year history.
"They both recognize there was a mistake," Chelsea manager Rafa Benitez said. "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."
On the club's official Twitter feed, Chelsea said football had "gone mad."
"Hazard is sent off for kicking the ball under a ball boy attempting to smother the ball rather than return it," Chelsea's tweet read.
Swansea manager Michael Laudrup said Hazard would regret his actions.
"The ball boy should have let the ball go, but he was pushed first and then he kicks him," Laudrup said. "As a player when you are behind, your pulse is very high and sometimes you say and do things. You overreact. But there are some things you cannot do."
Hazard is in his first season with Chelsea after joining the English club from Lille for a 32 million pound ($51 million) transfer fee