The Tigers have been in superb form on their own turf under Marco Silva but suffered a shock 2-0 defeat that was compounded by Paul Clement's men's 1-0 triumph over the Toffees in the evening fixture.
Swansea are now out of the bottom three and a point ahead of Hull with two games to play.
Here is your complete round-up of the goings on in England's top tier this weekend.
PREMIER LEAGUE SCORES
|Manchester City||5-0||Crystal Palace|
PREMIER LEAGUE TABLE
Chelsea will have the chance to extend their lead at the top to seven points on Monday night against Middlesbrough, and are now two wins away from the title. At the other end, all the teams on 41 points or more have now guaranteed survival.
Burnley and Watford are close to confirming their Premier League status for another year, but Crystal Palace remain in the mix. They face Hull next weekend in a match that will have massive ramifications at the bottom of the table.
HUGE BOOST FOR SWANSEA
Swansea are out of the bottom three.
There are two more matches to play and undoubtedly more ups and downs to come for both the Swans and Hull, but the Welsh club's fate is now back in their own hands.
They cannot have expected, given their recent excellence at the KCOM Stadium, that Hull would lose to bottom side Sunderland. The job for Swansea on this afternoon in south Wales was simply to stay in touch with the Tigers.
They did their job courtesy of a Fernando Llorente goal and are now a point ahead of them going into their own match against the Black Cats - away from home - next weekend. Hull have to go to Crystal Palace, themselves looking to ensure they stay out of relegation trouble, before hosting Tottenham.
Before Swansea's win over Stoke, Hull looked to have all the momentum. Seven points and three games later and everything has changed.
ALSO OF NOTE...
Leicester City have now played 10 league games under Craig Shakespeare and collected 22 points. That is an 83-point pace over a full Premier League campaign.
They are extremely unlikely to keep that up over the longer term, of course, but it has served to demonstrate that the Foxes' flirtation with relegation zone earlier this season was a club underachieving rather than one being brought back down to earth after their title triumph.
Leicester have a tried and trusted defensive group, a star in Riyad Mahrez and goals in Jamie Vardy, while Wilfred Ndidi has solved their biggest pre-Shakespeare problem of the way the team lost its identity without N'Golo Kante in midfield.
It will be fascinating to see how Leicester fare next season with Shakespeare - surely given the job permanently - at the helm. Having relied so heavily on a specific starting XI for two seasons in a row, they will be challenged with adding to and strengthening their group while not losing their magic formula.