Carlos Carvalhal's men made most of the running at the Liberty Stadium but Kyle Naughton's unfortunate own goal in the 43rd minute left them chasing the game.
That was something an Everton side grasping for any kind of fluency allowed them to do and Jordan Ayew – back in the side following a three-match ban for his red card in last month's 0-0 draw with Huddersfield Town – crowned an enterprising display with a fine 71st-minute finish.
Victories for Huddersfield Town and Crystal Palace have dropped Swansea down to 17th but the all-important gap between themselves and third-bottom Southampton is five points with as many games remaining.
Everton missed out on the chance to go above Leicester City into eighth after making it consecutive draws on the back of last weekend's goalless Merseyside derby.
AYYYYYYYYYYYYEWWWWWWWW!!!!! GET IN!!!!! pic.twitter.com/HsexR8dJsp— Swansea City AFC (@SwansOfficial) April 14, 2018
The hosts were on the front foot during the opening exchanges, as a disjoined Everton showing did not chime with Sam Allardyce's confidence that he will be their manager next season.
Swansea's first clear opening arrived in the 21st minute, with Everton opened up on the counter-attack and Idrissa Gueye's slip allowed Luciano Narsingh to find Jordan Ayew.
Leighton Baines managed to deflect the Ghana forward's attempt behind and striker Cenk Tosun cleared Federico Fernandez's header off the line from the resulting corner.
Andre Ayew was sent through by his brother Jordan in the 26th minute but shot too close to Everton goalkeeper Jordan Pickford under close attention from Michael Keane.
Everton grew into the game after the half hour but could still consider themselves very fortunate to lead in the 43rd minute.
Seamus Coleman raided forward and Yannick Bolasie's half-volley from the right-back's cross drew a brilliant save from Lukasz Fabianski.
The Swansea keeper was similarly sharp in keeping out Bolasie's follow-up but that rebound struck Naughton and crossed the line before Alfie Mawson could hack clear.
Pickford showed assured handling to deny Tom Carroll from a long throw early in the second period and the England international did well to parry behind from Andre Ayew at close quarters in the 51st minute.
Carvalhal sent on Tammy Abraham in an attempt to sharpen his team's attack and fellow substitute Nathan Dyer sent a rasping drive just over with 20 minutes remaining.
Tosun, largely starved of service in attack, then chipped in with another goalline clearance from Mawson but Everton could not fully clear the danger and Swansea pounced.
Whether Carroll meant to pick out Jordan Ayew with the outside of his boot was debatable but there was no doubt about the forward's unerring strike across Pickford.
Everton sprung from their slumber, with Coleman slipping a pass to Tosun down the right channel and continuing his run to slide a return ball against the crossbar.
Abraham lashed over as Everton's set-piece defending once again malfunctioned, while the pantomime jeers that greeted a late Wayne Rooney free-kick did much to summarise the efforts of Allardyce's side and his former England captain.
Key Opta stats:
- Swansea have only lost one of their last eight Premier League meetings with Everton (W2 D5), with the Swans unbeaten in their last four at Liberty Stadium (W1 D3).
- Swansea City have conceded 13 own goals in the Premier League since 2014-15; five more than any other side.
- Since keeping two clean sheets in December, Everton have conceded in each of their last seven Premier League games on the road.
- Jordan Ayew has scored in three of his last four Premier League games at Liberty Stadium, including of his last two.
- In fact, Ayew has scored seven goals this season in the Premier League – equalling his best-ever tally from 2015-16.
- Tom Carroll recorded his fourth Premier League assist of the season – double his tally from last season.