Leading 2-0 from the first leg, Swansea held out for a goalless draw at the Liberty Stadium to reach the first major final in their history, which will see the Welsh club meet fourth-tier Bradford City at Wembley on February 24.
It was a bad day all round for the visitors, who had Belgian winger Hazard sent off for lashing out in frustration at a ball boy with 10 minutes of the game remaining.
Needing to make a rapid start, Chelsea enjoyed plenty of possession in the opening quarter, with Hazard and Oscar covering just about every blade of grass in the Swansea half in pursuit of the gap that might prise open the hosts' back four.
Their slick passing might have brought greater reward in the seventh minute when Demba Ba appeared to be upended by Ben Davies in the Swansea box.
However, referee Chris Foy waved away the striker's claims and Swansea duly cleared from the ensuing corner.
Playing on the counter-attack was always likely to be Swansea's best mode of attack and it almost paid dividends 90 seconds after Ba's appeal.
A neat cross from the right created havoc at the far post, but Wayne Routledge's crisp volley was deflected over Petr Cech's crossbar for a corner.
Two minutes later, Routledge turned provider for leading scorer Michu.
The Spaniard, who set Swansea on their way at Stamford Bridge in the first leg, picked his spot, but Cech spread himself and pushed the ball to safety.
Knowing that a third goal would just about kill off the tie, Swansea broke forward in numbers whenever they had the ball.
Davies, fresh from his first senior goal for the club against Stoke City on Saturday, created the extra man on the left, while Angel Rangel did a similar job on the opposite flank.
In contrast, Chelsea did their best work in the middle third, where Hazard in particular continued to cause Swansea a headache with his prodigious movement. Yet for all Chelsea's quality, it was Swansea who looked the more comfortable.
Hazard's deflected shot for a corner in the 20th minute was Chelsea's first real opportunity.
However, it signalled a period during which Cahill had a goal-bound header cleared off the line by Rangel and Ramires, from 25 yards, was denied by goalkeeper Gerhard Tremmel.
Chelsea might have gone into the break in front had Ba have kept his shot down when an attempted clearance from Davies fell at his feet.
It was looking a little too easy for Swansea until Ba found himself enough space on the edge of the Swansea box, five minutes after the interval.
His shot looked set for Tremmel's far corner, but ended up a yard wide.
Mata thought he had given Chelsea a lifeline 17 minutes from time, but his smart shot was dealt with by the commanding Tremmel.
Then came Hazard's red card. The Belgian, Chelsea's star close-season signing from Lille, tried to wrestle the ball from a ball boy and appeared to kick out at the youngster, who was left in pain.
The incident, which saw Hazard shown a straight red card, overshadowed the game itself, but Swansea were good value for the draw and deservedly take their place in the final.
For Chelsea, Hazard's moment of fury made it even more of a night to forget.