Chelsea could not take advantage of long spells of possession as first Michu, then Danny Graham, scored to give the Welsh side the first leg lead.
The second leg takes place in a fortnight’s time, with Swansea on the brink of their first major cup final.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.