A miserable night for the European champions at Liberty Stadium was sealed when Hazard was shown a red card for kicking the ball boy in the ribs while attempting to retrieve the ball in the 79th minute, with Chelsea needing two goals to force extra time in the second leg of the semifinal.
The match finished 0-0, with Swansea winning 2-0 on aggregate and advancing to its first major cup final. The team will play fourth-tier Bradford City in an unlikely lineup for the showpiece at Wembley Stadium on Feb. 24.
Hazard and the 17-year-old ball boy, who appeared to be refusing to return the ball to the Belgium international to waste time, have apologized to each other. They were both spoken to by police and Swansea said no charges will be pressed against Hazard.
"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," Hazard said. "I apologize."
Hazard's actions overshadowed one of the biggest nights in the 100-year history of Swansea, a Welsh club which plays in the English Premier League and is coached by Michael Laudrup — one of Europe's great players from the 1980s.
"The ball boy should have let the ball go, but he was pushed first and then he kicks him," said Laudrup, a former Denmark, Real Madrid and Barcelona forward. "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."
Protecting a comfortable lead from the first leg at Stamford Bridge two weeks ago, Swansea defended stubbornly and conceded few chances to Chelsea's star-studded team.
A match bereft of chances and action came to life when Hazard ran up to reclaim the ball which had gone over the goal line, bounced off an advertising board and been collected by the ball boy.
The youth fell to the ground, holding the ball, which Hazard tried to pull the ball free with both hands. He then appeared to try to kick the ball underneath the ball boy but made contact with him.
Hazard was sent off by referee Chris Foy, sparking protest by Chelsea's players, and television replays showed the ball boy being led off along the edge of the pitch, clutching his ribs. It isn't thought that he resumed his duties.
"Has football gone mad?" Chelsea said on its official Twitter feed. "Hazard is sent off for kicking the ball under a ball boy attempting to smother the ball rather than return it."
Chelsea manager Rafa Benitez said Hazard met the ball boy in the changing rooms.
"They both recognize there was a mistake," 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.
"We can analyse if for half an hour but we know that both are wrong."
A fifth opportunity for silverware slipped by for Chelsea, whose defence of the Champions League ended in the group stage in December. With the team 11 points behind Manchester United in the Premier League, there looks to be only the FA Cup and Europa League to aim for this season.
Also Wednesday, Arsenal revived its hopes of finishing in the top four of the Premier League and qualifying for the Champions League by thrashing West Ham 5-1, with a scoring spree at the start of the second half doing the damage.
Tied at 1-1 at halftime, Arsene Wenger's side scored four times in 11 minutes at the start of the second half. Olivier Giroud, Santi Cazorla and Theo Walcott all grabbed goals, adding to Lukas Podolski's equalizer in the 24th minute.
Arsenal stayed sixth but moved one point behind fifth-placed Everton. Tottenham, which occupies the fourth and final Champions League spot, is four points ahead of Arsenal.