A fourth match without defeat for rejuvenated Swansea - their best such run of the campaign - leaves them within touching distance of safety, knowing that their two direct rivals for relegation face off at Selhurst Park on Sunday afternoon.
Paul Clement, boosted by an unchanged starting lineup for this match, saw his side take the lead through their first opening of the afternoon - Llorente getting to the ball ahead of Jordan Pickford with his head to open the scoring.
It was a rare goalkeeping error from the Black Cats' key asset, who came well off his line but failed to get anything on Gylfi Sigurdsson's inch-perfect delivery.
Swansea were looking very comfortable in possession, helped by an extra body in midfield, and they nearly added a second when Sigurdsson cleared the wall with his free kick but was denied routinely enough by Pickford.
Swansea failed to really build on a dominant start to the afternoon but they did bag themselves a second just before the interval, finding a way through thanks to a bullet strike from Naughton at the end of a Ki Sung-yueng through-ball - his first goal in 140 Premier League outings.
The hosts, now winless in 10 home matches to see out the season, had a penalty shout rejected just short of the hour after Sebastian Larsson's free kick hit Sigurdsson's arm and looped into the hands of Fabianski.
Moyes looked to match his side up man for man with the opposition in the second half, meaning that the teams essentially cancelled each other out in a flat conclusion to the afternoon.
Wahbi Khazri, one of Sunderland's first-half replacements, hammered wide from a volley and then gave Fabianski something to think about with a free kick from distance.
Fabianski could only parry into the path of Billy Jones, who somehow turned the rebound wide of goal, and lively Tunisian Khazri was to be thwarted again by the Swans stopper - previously without an away clean sheet since the opening day - late on.
Khazri was fortunate to avoid seeing red in the closing stages following a stamp on Sigurdsson, with referee Andre Marriner instead opting to show a yellow, but the attacking midfielder failed to get the better of Fabianski from his final attempt of another disappointing afternoon for the hosts.