Fernando Llorente fired Swansea out of the Premier League's bottom three as they beat Everton 1-0 on Saturday evening.
The Welsh side needed three points to leapfrog Hull City into 17th place after the Tigers lost 2-0 to already-relegated Sunderland earlier in the day.
And that is exactly what happened during a nervy encounter at the Liberty Stadium as Llorente nodded home from a Jordan Ayew cross in the 29th minute to seal a vital win for the Swans.
They now have 35 points - one more than Hull - while Everton missed the chance to go into the top six above Arsenal and remain in seventh spot.
Swansea made one change from the side that drew with Man Utd last weekend as Martin Olsson replaced Stephen Kingsley at left-back, while Ronald Koeman made two alterations from the 3-0 loss to Chelsea, although neither could muster much in the way of chances during the first 25 minutes.
The home side had a penalty shout in the 28th minute, with Gylfi Sigurdsson going to ground just inside the box after an apparent foul from Gareth Barry but referee Martin Atkinson waved away the claims. The Iceland international thumped the pitch in fury, but he was not left angry for long as Llorente netted a few seconds later.
Ayew put in some fantastic work down the right-hand side, digging the ball out under pressure from Barry to fire a cross into the box that met the head of the Spanish striker at the back post, rising above marker Phil Jagielka to score his 13th goal of the season and crucially put Swansea ahead.
It spurred on the hosts and they could have nabbed a quick second in the 33rd minute as Sigurdsson's cross from the right flank fell to the feet of Alfie Mawson at the back post. He found himself with plenty of space to shoot, but a great interception from Mason Holgate saw him turn the ball wide.
Koeman made a change at the break as Ross Barkely replaced Dominic Calvert-Lewin and he had an almost instant impact as Everton started the half strongly. He pumped a free-kick into the box, but Jagielka could not direct his header on target before Llorente cleared a rebounded cross into the box from Barry.
Ten minutes into the second period and Swansea could have doubled their lead. A poor headed clearance from Jagielka saw the ball fall to the feet of Ayew and he took aim, but his effort smashed against the woodwork and went wide.
Despite an improved display from Everton, who were seeing plenty of the ball, it was Swansea that continued to have the better chances. Sigurdsson delivered successive corners just before the 70-minute mark, with the first meeting the head of Federico Fernandez but ricocheting off the shoulder of Barkley before another Llorente header was well collected by Maarten Stekelenburg.
The hosts went close again in the 78th minute as they looked to secure their victory, producing a fine counter-attack that saw Tom Carroll skip away down the left-hand side before firing a cross into the box for the incoming Leroy Fer, but he was denied by the foot of Stekelenburg who pushed his effort onto the post and ultimately wide.
Everton were camped in the Swansea half in the final 10 minutes as the game drew to a nervous conclusion for the home fans. Barkley whipped a free-kick into the box with three minutes of normal time to play, which was flicked on by Kevin Mirallas at the near post for the head of Ashley Williams but it just evaded the former Swans defender before going wide.
Ultimately, they were unable to find their way past a determined Swansea side as the fans roared them home in added time, with scenes of celebration as the full-time whistle was blown.
Player ratings
Swansea: Fabianski (6), Naughton (7), Fernandez (8), Mawson (9), Olsson (7), Ki (7), Britton (7), Carroll (7), Sigurdsson (8), Ayew (7), Llorente (8).
Subs used: Cork (7), Fer (7), Baston (n/a).
Everton: Stekelenburg (6), Holgate (7), Jagielka (7), Williams (7), Baines (7), Gueye (7), Barry (7), Davies (6), Calvert-Lewin (6), Lukaku (7), Mirallas (7).
Subs used: Barkley (7), Valencia (6), Kenny (6).
Man of the match: Alfie Mawson.