Chelsea defender Branislav Ivanovic was to blame for both goals at Stamford Bridge. His poor control in the 39th minute allowed prolific forward Michu to curl home his 16th goal of the season, before a lazy back-pass was seized upon emphatically by Danny Graham in injury time.
With its attacking options, Chelsea cannot be discounted in the second leg on Jan. 23, but Swansea is in full control — especially with Michu in its ranks.
"It was the bargain of the season," said Swansea manager Michael Laudrup, who signed Michu for just 2 million pounds ($3.2 million) from Spanish side Rayo Vallecano in the off-season. "Only a few clubs can afford to buy him (now)."
With fourth-tier club Bradford holding a shock 3-1 lead over Aston Villa halfway through the other semifinal, it could be two of the most unforeseen finalists ever in the competition.
"There is still a mountain to climb," Laudrup said. "Chelsea have so much quality."
The League Cup would have been way down the list of priorities for Chelsea at the start of the season, but four opportunities for silverware have already been squandered — including an unprecedented elimination at the Champions League group stage as holders. Challenging for the Premier League title is also unlikely, given the team is 14 points behind leader Manchester United.
This game, therefore, meant a lot to the London club. But the recalled Fernando Torres toiled up front and only three genuine chances were created — all fired straight at goalkeeper Gerhard Tremmel by Ramires, Juan Mata and David Luiz. Demba Ba had a goal disallowed just before the final whistle, which was greeted with jeers and boos by the home crowd.
Most of them were directed at interim manager Rafael Benitez, still as unpopular as ever six weeks into his tenure, but the whole club seems in disarray. Even chairman Bruce Buck was booed before kickoff as he came onto the pitch to present an award.
"We made two mistakes and we paid for them," Benitez said. "If you play this game 10 times, you win nine of them."
Esteemed managers such as Manchester United's Alex Ferguson and Arsenal's Arsene Wenger have acknowledged they had barely heard of Michu when he arrived at Swansea, but they know all about him now.
His languid style is similar to that of Dimitar Berbatov, but he bursts into life in and around the penalty area and has a deadly finish, displayed when he put Swansea ahead from the edge of the area after Ivanovic had been robbed by Jonathan de Guzman.
"What he has done in terms of goals is incredible," Laudrup said. "Today — one chance, one goal."
Benitez answered the crowd's fevered calls to bring on Frank Lampard and Demba Ba, and the latter — a recent signing from Newcastle — did more in 10 minutes than Torres did in 80.
Making his presence felt immediately, he had a goal disallowed for offside, was booked for diving after colliding with Tremmel in what looked a harsh decision, and twice headed just off target.
Swansea's defence — marshalled superbly by centre back Ashley Williams — held firm and in the second minute of injury time, Ivanovic failed to spot Graham lurking behind him as he passed the ball back to Turnbull. The substitute pounced to round the 'keeper and slot home.
"We have had some historic results in the first half of the season," Laudrup said, "but to win at the European champions is something very special."
Swansea has won a glut of Welsh Cups but no titles in English football.