Llorente scored the game’s only goal, towering over Phil Jagielka at the far post to head home from Jordan Ayew’s cross.
First half dull bar a manic five minutes
News that Hull had lost to already relegated Sunderland will have reached the Swansea players before kick-off, and they will have known that if they win out, they will retain Premier League status for next season.
Everton were the better side in a very quiet opening 25 minutes, but Llorente was able to send the home fans into raptures with the game’s first shot on target.
Ayew lofted the ball to the back post from a tight area which was begging Llorente to meet with a header, and he was able to beat Jagielka and send Swansea 1-0 up.
Just moments earlier Swansea were claiming for a penalty, after Gylfi Sigurdsson went down in the box under pressure from Gareth Barry but referee Martin Atkinson waved the protests away.
Alfie Mawson then had a chance to double the lead as Swansea really took control, the ball fell to the defender at the far post but some sublime intervention from Mason Holgate was needed to block Mawson’s close-range strike.
Martin Olsson then had a chance, after Ayew managed to stop the ball from going out of play, he fed the full-back in on goal but his shot was blocked by Barry.
Everton have never lost a league match away to the Swans, winning eight of the 11 fixtures but that record was in doubt as the Blues went into the half-time break 1-0 down.
Huge second half from the Welsh side
Ronald Koeman subbed on Ross Barkley at half-time, as Everton looked for some more control in the final third.
Everton did look a lot better after the restart, immediately putting Swansea under pressure from a Barkley free-kick but they couldn’t fashion a shot from it as the Liberty Stadium collectively felt a lot more tense.
The post then denied Swansea a second goal, as the ball ping-ponged around the Everton area before falling to Ayew who struck the ball with his shin, hitting the post.
Ayew is still looking for his first Swansea goal, but looks a much better player in a two-man strike-force with Llorente as opposed to on his own or out wide, where he featured before the change of shape to a diamond.
Lukasz Fabianski was called into action for the first real time after an hour, going down to his right to stop Romelu Lukaku’s strike from range.
Everton started to up the ante with 20 minutes to go, as there was some pinball in the Swansea area which led to a chance for Kevin Mirallas but the ball was eventually cleared away.
Substitute Leroy Fer nearly scored moments after coming onto the pitch, he got onto Olsson’s cross and hit his shot first-time but Maarten Stekelenburg was on-hand to make a stunning save to keep Everton in the game.
Lukaku rippled the side-netting after ghosting past Olsson, cutting inside but aimed to go near-post as opposed to the far-post as we entered the final 10 minutes of the game.
Mirallas took the ball off of Ashley Williams’ head with five minutes to go, getting on the end of a free-kick and his effort flew a few yards wide of the post.
Not only will manager Paul Clement be happy with the result, today was probably the best Swansea have defended all season. Mawson looked like a brick wall at times.
All Swansea need to do to stay up now is win their two remaining games. Easier said than done, but at least their safety is in their own hands.
Swansea’s last two remaining Premier League games are away to Sunderland and then they host West Brom on the final day.