Ayew, back from a three-match ban, struck his 11th goal of the season after 71 minutes just when Everton appeared on course for a smash-and-grab win at the Liberty Stadium.
He's back on Saturday.
But the draw took Swansea five points clear of Southampton, which occupies the final relegation place after losing to Chelsea 3-2.
From the resulting delivery, Federico Fernandez's header was kept off the line by Cenk Tosun before Carroll fired over from an acute angle.
Seamus Coleman found Yannick Bolasie unmarked, but Lukasz Fabianski was equal to the winger's drive as well as Gueye's follow-up effort.
Trump Orders 'Sustained' Strikes On Syria Until Stops Using Chemical Weapons
Four RAF Tornado jets, operating from Akrotiri, Cyprus, each delivered a "Storm Shadow" missile at a target 15 miles west of Homs. The US has acted to punish the Assad government for using chemical weapons such as sarin and chlorine gas on Syrian civilians.
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.
Everton grew into the game after the half hour mark but could still consider themselves very fortunate to lead in the 43rd minute.
Swansea responded at the start of the second half with Carroll and Andre Ayew testing Pickford.
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.
Then, at the other end, substitute Tammy Abraham blasted over from Jordan Ayew's knockdown.
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.