Relegated Sunderland threatened to drag Hull City with them through the trap door after a surprise 2-0 victory on Saturday put a huge dent in their opponents' survival hopes.
Four consecutive home wins had put Hull's fate firmly in their own hands but, given the chance to pull five points clear of third-from-bottom Swansea City, they suffered an attack of the jitters that could prove fatal.
Second-half goals by Billy Jones and Jermain Defoe earned bottom club Sunderland a first win in 11 games and condemned Hull to a first home defeat under Marco Silva.
Should Swansea beat Everton later on Saturday, Hull would slide back into the bottom three with two games remaining.
Champions Leicester City finally assured their safety with a 3-0 victory at home to Watford, while Stoke City and Bournemouth will be playing top-flight football next season after a 2-2 draw on the south coast.
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 also drew 2-2 at home to West Bromwich Albion.
Sam Vokes scored both Burnley goals.
Manchester City were the only top-six club in action on Saturday and took advantage with a 5-0 thrashing of Crystal Palace who themselves are not safe from the drop after a third consecutive defeat left them in 16th spot.
After two consecutive draws, City needed a win to bolster their top-four prospects and delivered a dazzling display of attacking football to move above Liverpool into third spot.
City never looked back after David Silva's 50th goal for the club in the second minute although it was not until the second half that the floodgates opened with Vincent Kompany, Kevin de Bruyne, Raheem Sterling and Nicolas Otamendi all on target.
Victory moved City to 69 points, above Liverpool on goal difference. Fifth-placed Manchester United, who play sixth-placed Arsenal on Sunday, have 65.
Leaders Chelsea host Middlesbrough on Monday needing just six points from their last four games after second-placed Tottenham Hotspur went down 1-0 at West Ham United on Friday.
"To score five against this team is very good and it is an important result for the mini-league fight that we are in for places in the top four," Manchester City manager Pep Guardiola said.
"Our performance was high level. I'm pleased for our people we can finally enjoy a lot of goals here."
Palace are still anxiously looking over their shoulders at the teams below them although they remained six points above third-bottom Swansea before their late kickoff.
"All five goals were avoidable. We didn't make them produce their brilliant talent, we gifted them five goals. They punished us heavily," Palace manager Sam Allardyce said.
After flirting with relegation for most of the season champions Leicester City moved into the top half of the table with the victory over Watford.
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