The youngster had refused to return the ball to play quickly as it went behind the goal, and ended up laid on top of the ball as the Belgian came to confront him.
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After much deliberation, referee Chris Foy brandished the red card, and Hazard will face at least a three-match ban.
Chelsea's official Twitter account reacted incredulously to the decision, saying: "Has football gone mad? Hazard is sent off for kicking the ball under a ball boy attempting to smother the ball rather than return it."
Swans skipper Ashley Williams also commented on the incident after the game, blasting the Belgian's actions.
He told Sky Sports: "Demba Ba told me that the boy held onto it but I saw him kick him in the ribs and you can't do that."
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The post read: "The king of all ball boys is back making his final appearance #needed #for #timewasting"
Former Premier League referee Dermot Gallagher told the BBC: "Chris Foy has quite rightly sent him off and what he will do now is report Eden Hazard's violent conduct to the FA."
Chelsea's official Twitter account later apologised for their mid-game comment adding: "Apologies for earlier ballboy tweet. Hazard has now met with the ballboy and has said sorry."