The visitors pulled off a great result in the Capital One Cup semi-final with a goal in each half giving them a vital lead in the tie ahead of the second leg at the Liberty Stadium
Rafael Benitez selected a strong side for the tie, although opted to rotate his attacking options with Fernando Torres replacing Demba Ba and Oscar returning to the starting line-up.
Swansea, meanwhile, made five changes, most notably Michu coming in with Danny Graham on the bench.
The hosts dominated the early exchanges and took just 10 minutes to carve out a great chance. A flowing move saw David Luiz and Eden Hazard combine beautifully to release Ramires into the area. The Brazilian took his time, but could only poke his shot tamely goalwards with an effort that did not trouble Gerhard Tremmel.
The goalkeeper was called into action again soon after, however. More slick approach work from the Blues saw Hazard again surging forwards before freeing Mata but the Spain international's side-footed shot lacked power and flew straight at Tremmel.
Minutes later and Mata was at it again, a clever backheel by Oscar left the Spaniard clean through on goal but he could only drag his shot wide when well placed.
Yet Chelsea were made to rue their missed chances when Swansea took the lead seven minutes before the break.
A simple pass by goalkeeper Turnbull was miscontrolled by Ivanovic on the edge of the area, allowing De Guzman to chase the defender down. However, the loose ball instead broke to Michu who took a touch before bending it superbly past the goalkeeper to open the scoring with a sublime finish.
The Blues were stunned but Ivanovic almost made amends just minutes before the break. The Serbian cut inside from the left before fizzing a shot goalwards which a diving Tremmel managed to push around the post.
The chances continued to come for Chelsea after the break with David Luiz wasting two free kicks before slamming a fierce shot straight at Tremmel, just shy of the hour mark.
With time running out Benitez sent on Frank Lampard for Ramires, much to the delight of the home fans. However, the midfielder will have been relieved to see the assistant referee's flag deny Michu a second, for offside, shortly after his arrival.
New signing Ba was the next to arrive, for a home debut, with 10 minutes to go in place of Fernando Torres and the former Newcastle man had two great chances within minutes of his arrival but could not find the back of the net, heading straight at Tremmel before heading another opportunity over.
His night then proceeded to get even worse as after being sent clean through on goal, he tangled with Tremmel, and with the crowd howling for a penalty, inside found himself booked for diving.
Again, the misses were to prove costly as Swansea doubled their lead in stoppage time, again through a defensive mistake by Chelsea. For the second time in the match, Ivanovic was to blame. The defender's poor backpass to Turnbull was pounced on by substitute Graham who rounded the keeper before calmly slotting home.
Yet there was still time for more drama, with just seconds left on the clock, Ba smashed the ball home after good work by Lampard, only to be denied by the offside flag, to leave the Blues with a huge challenge ahead of them.