Eden Hazard was sent off for appearing to kick a ball boy late on as Rafa Benitez's side could not break down Michael Laudrup's team, which reached its first-ever major cup final
The night was a huge blow to Rafa Benitez who also saw Eden Hazard sent off for appearing to kick a ball boy, and the result leaves the FA Cup and Europa League as the Stamford Bridge club's only realistic chances of winning a trophy.
The Swans, meanwhile, can look forward to their first ever appearance in a major cup final, and could become the first Welsh winners of the league cup in the process in a terrific first season in charge for Laudrup.
Controversy ensued after just five minutes in a lively opening - Demba Ba (preferred to Fernando Torres) went down in the area as Ben Davies hung out a leg, but the striker's theatrical reaction failed to persuade referee Chris Foy.
Swansea continued the entertaining start with Michu stinging Petr Cech's hands with a fine 16-yard drive after a clever through-ball by Wayne Routledge, with the hosts showing no fear in defending their two-goal cushion.
Gerhard Tremmel scooped up Hazard's low effort on 20 minutes with Ba lurking for the rebound and did well to hold Ramires' strong drive three minutes later as Chelsea began to threaten the hosts.
Juan Mata couldn't quite control a Ba pass well enough to feed Oscar who would've had a simple finish from 12 yards, but the flair of the visitors' forwards began to shine through despite the confident passing of Laudrup's side.
While many would have assumed Chelsea to push on in the second half in search of goals, the visiting fans were left frustrated with the likes of Mata and Ramires seeing shots closed down by a determined display from the hosts.
Chelsea upped the tempo as Mata unleashed a fine 20-yard drive with 18 minutes to go that Tremmel expertly held on to, while Torres was introduced with 13 minutes to try and rescue the game.
However, all hopes of a comeback faded after Hazard received his marching orders with 10 minutes to go after bizarrely kicking a ball from underneath a ball boy, who appeared to be lying on the ball to waste time.
Swansea ended up looking the likelier side to score late on as Ki and Nathan Dyer wasted opportunities, but the Swans will not care about failing to score as they have a dream date at Wembley to look forward to.