The Swans maintained their impressive run of form against the Toffees, making it six without defeat in this fixture thanks to Llorente's close-range header 29 minutes in.
A shock defeat for Hull City at home to relegated Sunderland earlier in the day left the door open for Swansea to climb out of the dropzone, with the Welsh outfit's victory taking them a point above the Tigers with two games to play.
A cagey opening quarter to the game saw both sides struggle to create any real sort of openings, but when Llorente's big moment arrived he made no mistake by heading Jordan Ayew's cross past Maarten Stekelenburg.
Ayew was denied by the foot of the post from his volleyed attempt as the relegation-threatened home side pressed for another, while up the other end Romelu Lukaku flashed the ball across the face of goal.
Lukaku tried his own luck 65 minutes in when testing Lukasz Fabianski with a firm shot on goal, but it was proving to be a testing evening for the Belgian forward.
A chance came and went for matchwinner Llorente to double his tally, rising highest to meet Gylfi Sigurdsson's corner but failing to find a way past Stekelenburg on this occasion.
Everton's best chance of rescuing a point arrived in the final 10 minutes, with Kevin Mirallas heading wide from a Ross Barkley free kick, while Mawson was well positioned to clear Lukaku's shot off the line.
Swansea were able to see out the closing seconds to make sure of the win, ensuring that Everton - winless on their travels since January - remain rooted in seventh place.