Hull, who earlier lost 2-0 at home to relegated Sunderland, drop into the bottom three, while this result also means Middlesbrough will be relegated to the Championship if they lose to leaders Chelsea at Stamford Bridge on Monday (20:00 BST).
Fernando Llorente will be available for Swansea as they look to make a final push for Premier League survival when they host Everton on Saturday (today).
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Two defeats were followed by a win apiece before battling draws last weekend, Hull holding Southampton and Swansea leaving Manchester United with a share of the spoils.
"We played against a really good team".
It lifted Swansea one point clear of Hull, who have slipped into the bottom three, with a trip to Sunderland and a home game against West Bromwich Albion to come.
"I think it is definitely going to go all the way down to the last game", he told the media.
Worse still, the Everton has failed to score on five separate occasions at home, and the club will be hoping that their attacking juggernaut Romelu Lukaku will rediscover his shooting boots, when the side locks horns with Swansea AFC this Saturday.
Their 5-0 win at home to Crystal Palace dislodged Liverpool and leaves them still in with a very slim chance to finish in second ahead of Tottenham.
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But Swansea fans booing Ashley Williams - someone who played over 350 games in leading the club up the football pyramid and skippered Wales to the semi-finals of Euro 2016 - surely ranks as the most crass jeering of an old favourite this season.
It felt, though, to be a decisive moment as any side battling relegation can not afford to squander such gilt-edged opportunities.
But Llorente was simply relieved after spurning a golden opportunity to grab a last-minute victor in last week's 1-1 draw at Manchester United.
We needed to win, we don't deny that.
The Premier League leaders took their time to break down Everton, but eventually ran out as 3-0 winners.
This Swansea side has far more top flight know how than their counterparts at Hull and the ability to grind out results in high-pressure games will be crucial at this time of the season. We have all handled pressure games in the past, we delivered in "must-win" games in the past against Boro, West Ham, Watford.
City dominated possession and pounded the Palace goal - Sane was thwarted three times by goalkeeper Wayne Hennessey - but still gave its opponent space at the other end, with Willy Caballero saving well from Christian Benteke's downward header late in the first half.
Three of Swansea's last four goals against Everton in the Premier League have been from the penalty spot, two of which were converted by Gylfi Sigurdsson.
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