However, Swansea, now with 33 points from as many games, are five points clear of third-bottom Southampton - with five games left of the 2017-18 English top-flight season.
From the resulting corner, Federico Fernández met Carroll's delivery from the right and powered a goalbound header but Cenk Tosun manning the near post headed off the line. Ayew's lovely first-time pass released his brother Andre through on goal but the forward's shot was tame and straight at Jordan Pickford.
Unfortunately for him, the second bounced off the head of Naughton and back towards the goal and though Alfie Mawson cleared the ball it was deemed to have crossed the line.
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 finish across Pickford.
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.
Everton's Jordan Pickford looks dejected after conceding their first goal to Swansea City during their Premier League match at Liberty Stadium, Swansea, April 14, 2018.
Supporters were unhappy pre-match after hearing that Idrissa Gueye was selected alongside Morgan Schneiderlin as a defensive midfield unit, questioning the need for such a negative line-up for against a team sat 15th in the table - just five points above the relegation zone before the start of play.
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