With Chelsea needing two goals at Liberty Stadium to force extra time, the ball bounced off an advertising board behind the goal line in the 79th minute and a 17-year-old ballboy in a Swansea tracksuit slowly walked over to pick it up.
Hazard ran up to reclaim the ball and the teen fell over, lying face down with the ball under him. Hazard, thinking the ballboy was trying to waste time to aid his team, attempted to pull the ball free 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 ballboy, 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 manager Rafa Benitez said the ballboy and Hazard had apologized to each other.
"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.''
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.
The match finished 0-0 and Swansea advanced 2-0 on aggregate, setting up an unlikely final against fourth-tier Bradford City at Wembley Stadium on Feb. 24. It will be the first major final in Swansea's 100-year history.
Swansea manager Michael Laudrup said Hazard would regret his actions.
"The ballboy 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 from Lille for 32 million pounds ($51 million).