Only the bottom three teams in the league, possesses a worse home record than Swansea AFC this season.
Swansea have taken seven points from their last three Premier League games after picking up just one in the six before that.
However, their defensive positioning remained very good and Everton, despite bringing on Ross Barkley, couldn't find a way to create any real chances.
Player ratingsSwansea: Lukasz Fabianski: 7, Kyle Naughton: 7, Federico Fernandez: 8, Alfie Mawson: 8, Martin Olsson: 8, Leon Britton: 8, Tom Carroll: 7, Ki Sung-yueng: 7, Gylfi Sigurdsson: 7, Jordan Ayew: 9, Fernando Llorente: 8. The way they got behind the players was superb.
Marco Silva's 17th-placed Hull have a chance to pull further clear from the drop zone against basement side Sunderland, before Swansea against seventh-placed Everton later on Saturday.
It was advantage Swansea on Saturday in the battle for survival after a 1-0 success over Everton in the Premier League. Raheem Sterling and Nicolas Otamendi, again from a free kick by De Bruyne, added late goals as Palace was swept aside, leaving the visitors still to definitely secure their top-flight status for another season.
The English premier league approaches the final stages of another captivating season this Saturday (May 6th), with only a handful of matches left to play, as league rivals Swansea AFC and Everton FC square off at the Liberty Stadium, with the kick-off commencing at (18:30 CET/17:30 GMT).
With Middlesbrough close to joining Sunderland in the Championship next season, the favourites to be the third relegated side are either Hull or Swansea.
Everton played with much more urgency after the break and as a result, Swansea lost their control of the game.
Sam Vokes scored both Burnley goals.
Not only will manager Paul Clement be happy with the result, today was probably the best Swansea have defended all season.
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.
Nonetheless, his goals during this campaign have helped Koeman's team into Europe but fans will be hoping their key man is still at the club next term.
The Baggies responded by scoring twice in 12 minutes - through Salomon Rondon and Craig Dawson - having previously gone almost nine hours without a goal.