Hazard was shown a straight red card by referee Chris Foy in the 80th minute of Wednesday's semi-final second leg after attempting to kick the ball from beneath a ball boy who was lying on top of it, apparently deliberately.
The pair later apologised to each other, but Hazard could still face charges from the Football Association after receiving the first red card of his career.
The boy was identified in media reports as Charlie Morgan, the 17-year-old son of Swansea director Martin Morgan.
In a message on his Twitter account written before the game, Charlie Morgan appeared to joke that he would try to waste time in a bid to help the home side protect their 2-0 aggregate lead.
"The king of all ball boys is back making his final appearance #needed #for #timewasting," he wrote.
Stoke City striker Michael Owen defended Hazard's actions on Twitter, writing: "Both the kid and Hazard were in the wrong.
"Not having (accepting that) Hazard tried to hurt him though. He just tried to toe poke it out of his grip."
Tottenham Hotspur winger Gareth Bale also sided with the Belgian international, tweeting: "Unbelievable decision by the referee to send Hazard off but congrats to Swansea."
Hazard, 22, has made an impressive start to life at Stamford Bridge, scoring eight goals and laying on 12 assists in 37 games, and one of his Chelsea predecessors, Pat Nevin, believes he was more sinned against than sinning at Swansea.
"I would have kicked the ball out from underneath the ball boy if he had been lying like that, 100 per cent," Nevin told BBC Radio 5 Live.
The former Chelsea winger also branded Charlie Morgan's behaviour "disgraceful".
"I was very, very disappointed by the way the ball boy acted - and I say 'acted'," he said.
"He must have been watching footballers, the way he rolled around and pretended to be more injured."
However, the chairman of players' union the Professional Footballers' Association (PFA), Gordon Taylor, said Hazard's behaviour was "unacceptable".
"You can't take the law into your own hands," Taylor told BBC Sport.
"You can understand the frustration of wanting to get the ball back, but his actions were unacceptable and the referee had no alternative."
Swansea will face fourth-tier Bradford City in the tournament final on February 24.