The Swans recovered from their unfortunate setback at the end of a first half that they dominated to claim a deserved draw from the contest.
A point apiece denied Swansea the chance to rack up five successive home wins in the top flight for the first time since 1981, while Everton are still awaiting their first back-to-back away victories since September 2016.
Seeking a first win in four matches, Swansea looked by far the better side prior to Everton's opener against the run of play.
Ki Sung-yeung had a couple of decent openings early on, sending one straight at Jordan Pickford and another high and wide, while Jordan Ayew controlled a pass in the box at the end of a counter but saw his shot hit Leighton Baines and trickle wide.
Jordan Ayew, back in the side today following a three-match ban, flicked the ball into the path of brother Andre for the Swans' next clear-cut chance, which the forward failed to convert from a one-on-one position.
Everton started to see a little more of the ball as the first half wore on, and finally had an effort on target 36 minutes in through a tame Yannick Bolasie shot.
Despite being second best up until that point, though, it was the Toffees who took a narrow lead into the break thanks to a bizarre own goal shortly before the interval.
Swansea were given a further blow for the start of the second half as Luciano Narsingh failed to shake off an injury, but it was the hosts who were continuing to ask all the questions.
Andre Ayew and Tom Carroll both had attempts kept out by Pickford, the former at the end of a slick passing move, and Narsingh's replacement Nathan Dyer clipped the roof of the net with his dipper soon after.
Following another spell of possession the Welsh outfit finally dragged themselves back on level terms, with top scorer Jordan Ayew controlling Carroll's pass and burying it past Pickford after Everton simply failed to clear their lines.
Chances came and went for both sides in the minutes that followed - Seamus Coleman hitting the crossbar with his volley from six yards and Tammy Abraham sweeping the ball over the target after Jordan Ayew headed a free kick perfectly into his path.
Both teams went in search of a winner late on but in the end had to settle for a share of the spoils, leaving Swansea five points clear of the relegation zone and moving Everton to within a point of eighth-placed Leicester City.
SWANSEA CITY (4-3-3): Fabianski; Naughton, Fernandez, Mawson, Olsson; Ki (Abraham 65'), King, Carroll; Narsingh (Dyer 46'), J.Ayew, A.Ayew
EVERTON (4-2-3-1): Pickford; Coleman, Keane, Jagielka, Baines; Schneiderlin, Gueye (Baningime 69'); Walcott, Rooney (Vlasic 87'), Bolasie (Funes Mori 76'); Tosun