The Swans were denied a crucial away win as Everton snatched a point via Seamus Coleman's 89th minute equaliser.
Bob Bradley's side looked to be heading for all three points at Goodison Park thanks to Gylfi Sigurdsson's first-half penalty, but in a second half in which they were forced to soak up a lot of Everton pressure, the honours were shared as Coleman headed into the top corner from inside the box.
It was a cruel end to a game in which the Swans put in one of their best performances of the campaign.
The point was not enough to stop Bradley's men from dropping to the bottom of the table, though, as Sunderland's win over Hull put them two points ahead.
This draw leaves the Swans on six points from 12 games - five points adrift of West Ham and Crystal Palace, who visit the Liberty next week.
Swans boss Bob Bradley made eight changes to the starting XI.
From the team that suffered a home defeat to Manchester United before the international break, four of those changes came at the back.
Federico Fernandez and Jordi Amat returned to the heart of defence as they made their first appearances since the narrow loss at Arsenal last month, while Wales international Neil Taylor and Kyle Naughton were named as full-backs.
In midfield, Jack Cork captained the side in the absence of Leon Britton - who missed out with a calf injury - while top scorer Leroy Fer back in the starting XI after coming off the bench against United.
The other change in midfield saw Jay Fulton make his second Premier League appearance of the campaign and his first since the home clash with Liverpool at the beginning of October.
Gambian winger Modou Barrow started on the wing while Gylfi Sigurdsson led the line.
On the bench, Nathan Dyer was included after recovering from an ankle injury while Oli McBurnie joined record-signing Borja as the attacking options at Bradley's disposal. Winger Jefferson Montero was not considered despite being given the go ahead to play with a cast after breaking his hand against United.
Everton, who were beaten 5-0 by Chelsea last time out, included ex-Swans skipper Ashley Williams in their line-up as he faced his former club for the first time since leaving the Liberty.
EVERTON: Maarten Stekelenburg, Leighton Baines, Ashley Williams, Phil Jagielka (capt, Enner Valencia, 83), Ross Barkley, Romelu Lukaku, Aaron Lennon (Gerard Deulofeu, 66), Yannick Bolasie, James McCarthy (Kevin Mirallas, 72), Idrissa Gueye, Seamus Coleman.
SUBS: Joel Robles, Tom Cleverley, Ramiro Funes Mori, Mason Holgate.
SWANS: Lukasz Fabianski, Kyle Naughton, Federico Fernandez, Jordi Amat, Neil Taylor, Jack Cork (capt), Jay Fulton (Ki Sung-Yueng, 87), Leroy Fer, Wayne Routledge, Modou Barrow (Nathan Dyer, 81), Gylfi Sigurdsson.
SUBS: Kristoffer Nordfeldt, Angel Rangel, Mike van der Hoorn, Borja, Oliver McBurnie.
After a relatively quiet opening 20 minutes, the Swans settled further into the game.
Lukasz Fabianski collected Romelu Lukaku's bouncing effort after his initial parry, while. at the other end, Leroy Fer wasted a decent opportunity when he skied his 20-yard effort.
The Swans were containing the hosts, and their shape seemed to thwart their attacking options.
Bob Bradley's side looked busy on the counter attack though, and when Modou Barrow was clattered by Ashley Williams, the Swans had a free-kick in a dangerous position.
Gylfi Sigurdsson stepped up, and his deflected effort was tipped over the crossbar by the diving Maarten Stekelenburg.
Then, from the corner, Jordi Amat rose highest only to direct his powerful header inches over.
Everton were probing though, and the best chance of the game so far fell to Lukaku, but he directed Yannick Bolasie's fine cross over from just six yards.
The lively Barkley fired wide on the turn from a Bolasie cross, while the former then saw his low free-kick spilled by Fabianski.
A minute later, the Swans took the lead.
With 41 minutes on the clock, Barrow touched on to Sigurdsson, who took a first touch into the box before feinting a shot and was then hauled down by Phil Jagielka.
Sigurdsson, as he did at this ground last season, fired home the penalty to send the Jack Army into elation.
There was still more drama to come, however, as Bolasie found himself in on goal with only Fabianski to beat, but the Polish keeper came out on top to deflect his effort wide.
That was the end of a thrilling first half, with the Swans protecting a precious lead.
HALF-TIME: Everton 0 Swans 1
The Swans were made to dig deep at soak up the waves of Everton pressure after the break.
In terms of chances, they were few and far between as the away defence held firm.
But for all the Swans' defensive effort, Everton found a way through.
After Sigurdsson had been denied by Stekelenburg late on, the hosts levelled in the 89th minute.
After the Swans failed to clear a cross, Seamus Coleman rose to nod perfectly into the top corner.
That wasn't it, for there were four minutes of added time to survive.
The Swans did hold on for a point despite the late pressure, though they slipped to bottom of the table as Sunderland beat Hull City.