The Swans will take on League Two's Bradford City in the final at Wembley on Sunday, February 24 after the Bantams stunned Aston Villa last night.
Tonight's match was a dogged affair, with little of note as Swansea nullified much of Chelsea's attacks. That however changed in the 80th minute when Eden Hazard was sent off for appearing to kick out at a ballboy who was lying on the ball, perhaps timewasting.
Below, Sports Mole pores over a night which saw Swansea reach their first ever major final in their centenary year.
On target 4
On target 5
Was the result fair?
Yes. Chelsea were expected to come out and possibly roast the Swans but that didn't materialise and Swansea actually had the best chances.
Michael Laudrup will be delighted with his side's resolute display. They have still only lost twice at home this season and their defence and midfield looked very cohesive and secure tonight. Michu missed a one-on-one chance in the first half before they could have nicked a win on the night on numerous occasions on the counter late on.
Pretty lifeless. The Blues didn't begin the match strongly and never really got going. The playmaking trio of Hazard, Oscar and Mata didn't produce much for themselves or Demba Ba and it was made worse when Hazard saw red in mad circumstances.
Sports Mole's man of the match
Ashley Williams: The Swansea captain has been talismanic at the back this season and he was on top form again tonight. Not only does he win almost all of the headers he goes up for, but he also reads the game very well. He made a number of interceptions to thwart Chelsea attacks tonight.
Should we give it to Hazard for kicking out at a ballboy? Oh yes. Of course, the young lad was out of order for not giving the ball back instantly but even though Hazard was trying to get the ball back, you can't do what he did.
Chris Foy was on top of the game until the 80th minute when he had to deal with the Hazard incident. In truth, he probably did the right thing in dismissing the Belgian but what a tough call.
Swansea: Michael Laudrup's side now have a six-day rest before a Premier League match at improving Sunderland on Tuesday.
Chelsea: The Blues, meanwhile, are in FA Cup action at the weekend. They travel to Brentford of League One at midday on Saturday for a fourth-round tie.