Jose Mourinho takes his Manchester United side to face Arsenal on Sunday with both teams now sat outside of the top four.
"We have to make sure now that we are absolutely focused to get a good result next weekend at Sunderland". The Toffee's are now ranked 6th in the league, and the club has big aspirations of European football next season. It looks to be down to three teams - Palace, Swansea or Hull - for the other relegation place. The Swans are now in the relegation zone, two points behind Hull and safety.
Hull's shock 2-0 defeat to already-relegated Sunderland had handed Swansea the ideal incentive before their Saturday evening kick-off.
"It is the nature of the league where anyone can beat anyone when firing to their potential".
John O'Shea flicked on a corner and Billy Jones launched into a diving header that flashed past Eldin Jakupovic. Ultimately there was big tension in some of the team.
"It's disappointing", Hull manager Marco Silva told the BBC.
"Staying calm is very important for us now", he said.
"Of course it came at a really important time for us in the game...so it was nice to get that goal and I think the performance deserved a point".
The mood among the Swansea supporters has been transformed since earlier in the season when the combination of the controversial change of ownership last summer and bad results on the pitch made the atmosphere at the Liberty Stadium toxic.
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Fernando Llorente's close-range header on the half-hour mark proved enough for the Swans, who now lead the Tigers by one point.
"When it comes to "squeaky bum time", it takes men to stand up and we did that".
Swansea City: Lukasz Fabianski, Kyle Naughton, Federico Fernandez, Alfie Mawson, Marin Olsson, Tom Carroll, Gylfi Sigurdsson, Leon Britton (capt) (Jack Cork 73), Ki Sung-Yeung (Leroy Fer 75), Tom Carroll, Fernando Llorente (Borja 87).
29' - GOAL! Swansea 1-0 Everton (Llorente).
Junior Stanislas equalised for Bournemouth in the 62nd minute before Stoke's Mame Biram Diouf bundled in his first goal for 12 months in the 73rd minute.
Stoke ended a 10-hour away goal drought in the 33rd minute when Bournemouth striker Lys Mousset headed the ball into his own goal but another own goal, from Ryan Shawcross late on, denied Stoke victory.
Burnley are not mathematically safe after they drew 2-2 at home to West Bromwich Albion with Sam Vokes scoring a brace for the hosts.
Midfielders Wilfred Ndidi, Riyad Mahrez, and Marc Albrighton scored the goals for the soon-to-be-deposed champions, who were close to the relegation zone when Ranieri departed and Craig Shakespeare came in as his replacement.