Demba Ba struck another priceless goal following his Champions League heroics to keep Chelsea firmly in the Barclays Premier League title race.
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Ba's 68th-minute strike finally broke 10-man Swansea's Liberty Stadium resistance for a 1-0 win that leaves Chelsea two points behind leaders Liverpool with four games remaining.
It followed the winner he struck against Paris St Germain five days ago that booked the Blues' place in this season's Champions League semi-finals.
Chelsea, who visit Anfield in a fortnight's time, were made to sweat and toil by the Swans following Chico Flores' 17th-minute sending off.
Flores was dismissed after picking up his second yellow card - his cautions came just over two minutes apart - but Swansea scrapped for everything before Ba struck and left Garry Monk's men just three points above the relegation zone.
Much of Chelsea's performance was hesitant, especially in attack, but Ba ultimately came up trumps when it really mattered - as he did at Stamford Bridge to dump PSG out of Europe on away goals - and it keeps the heat on Liverpool.
Swansea, in contrast, head to Newcastle next Saturday, knowing they are far from safe, especially following a weekend when two of the clubs below them - Fulham and Cardiff - both won.
Mourinho made five changes from the side that took on PSG, handing starts to Ba, Nemanja Matic, Ramires, Mohamad Salah and Andre Schurrle.
Swansea showed two switches from the side beaten 1-0 at Hull last weekend, with Nathan Dyer and Pablo Hernandez both starting. Jonathan De Guzman and Michu dropped out, with no place for Michu among the substitutes either.
A fast opening saw Chelsea goalkeeper Petr Cech save low down from Hernandez, before Salah missed a golden sixth-minute opportunity for the visitors when he side-footed wide from inside the penalty area.
Swansea enjoyed the lion's share of territory, and Cech was called on again to deflect a Bony header over the bar before Flores became an unwanted centre of attention.
He was cautioned for a foul on Willian, then he pulled down Schurrle just outside Swansea's box, and referee Phil Dowd brandished another yellow, leaving the home side down to 10 men.
Swans head coach Monk was left fuming by the decision - Dowd appeared to take a long time before he brandished a card - and he cut an animated figure on the touchline before a sense of order was restored and a reshuffle meant Jordi Armat going on for Hernandez.
Only two players had been sent off earlier during a game for two yellows cards in Premier League history - former Arsenal captain Tony Adams and Everton's Alex Clelland.
It meant Swansea were left with a mountain to climb, but despite their disadvantage in terms of personnel there was an assuredness about the Welsh club, exemplified by skipper Ashley Williams.
Chelsea displayed their trademark patience in attempts to break Swansea down, but keeper Michel Vorm had still to be seriously troubled as the game moved towards half-time.
Williams' first mistake saw him concede a needless corner five minutes before the break, but Branislav Ivanovic headed over after climbing highest among a melee of players.
And then Cech was forced into action, turning a powerful Dyer shot around his left-hand post as Swansea continue to be unfazed by the challenge they faced.
Schurrle was then cautioned for a lunge on Angel Rangel in the final notable action of a scoreless opening period.
Mourinho made a double substitution for the second half, sending on Oscar for Ramires and Samuel Eto'o instead of Schurrle.
Eto'o had a glorious chance to fire Chelsea ahead in the 58th minute, yet his miss from just 10 metres out epitomised the Blues' blunt attacking threat as Swansea remained on level terms.
And then a thrilling break out from a prolonged spell of defending almost produced the opening goal as Wayne Routledge's shot deflected off Chelsea skipper John Terry and bobbled wide.
But Ba finally destroyed Swansea's hopes of a point, leaving them still scrapping in the league's lower reaches and ensuring Chelsea retain hopes of title silverware as they survived a late Swans flurry that saw Cech pressed into action.