The Welsh club now lie just above the relegation zone, and have the edge over Hull City with just two games remaining.
The already-relegated Black Cats have picked up a paltry 14 points from 19 home games this term, while Swansea secured three vital points in the fight for survival to move four clear of Hull, who have two games to play.
Paul Clement's side sits just one place and one point above the relegation zone.
The Welsh club - bottom and four points adrift of safety in January when Paul Clement arrived as their third boss this season - are now four points clear of third-bottom Hull.
That's what you want from your striker.
An early injury to Jason Denayer saw Darron Gibson, disliked nearly as much as Moyes by sections of the crowd, booed onto the pitch.
Llorente scored the only goal in their win over Everton last weekend and Clement said: "He's been massive in recent games with big goals". Instead, we'll bank at least one side not scoring.
"The first half was strong from us and the goal before half-time gave us the confidence to see the game out". His angled drive - set up by a neat pass from former Black Cats loan player Ki Sung-yueng - ensured there would be no late nerves for his side. This time there should be no such collapse, and a win over Sunderland this Saturday could finally confirm Swansea's survival.
Short can not have failed to notice how many home fans had already departed by the time the second half kicked off.
The hosts ought to have had a penalty after 58 minutes with referee Andre Marriner inexplicably failing to spot Sigurdsson blocking Sebastian Larsson's free-kick with his outstretched arm.
Khazri was lucky not to suffer worse punishment when referee Andre Marriner booked him for what looked like a stamp on Sigurdsson. Perhaps galvanised by a sense of injustice Sunderland rallied a little and Khazri sent a volley whizzing fractionally off-target.
And, three minutes later, full-back Billy Jones somehow shot wide from close range after Fabianski had pushed substitute Wahbi Khazri's free-kick back into a unsafe area.
For opposite number David Moyes, it proved a dispiriting afternoon as fans once again turned on him. Sunderland have won half of their four home games versus the same opposition and going back to last season it's just three losses in nine such matches.
The Swansea players have covered the cost of 3,000 tickets for fans that are travelling for the crucial 700-mile trip.