Aaron Lennon gave the visitors a 41st minute lead with his second goal in Everton colours following his recent winner at QPR and manager Roberto Martinez looked on course for a successful return to his old home.
But Swansea equalised after 69 minutes when Seamus Coleman's inexplicable handball was spotted by referee Michael Oliver and Jonjo Shelvey scored with a firm spot-kick.
Everton had won on their three previous Premier League visits to the Liberty Stadium and Swansea had never beaten the Toffees in any competition before September's 3-0 victory in the Capital One Cup.
On the face of it eighth against 12th suggested a calm end-of-season affair but it was anything but as Oliver was tested throughout and did well to keep the full complement of players on the field.
Both camps will probably be content with a point as Swansea made it three games unbeaten and reached 47 points - the mark set in their first Premier League campaign in 2011-12 - while improving Everton have now gone four games without losing in the top flight.
Swansea started brightly with Bafetimbi Gomis in confident mood after three goals in his last two appearances.
Gomis was stopped by a last-gasp Leighton Baines challenge and then warmed Tim Howard's palms from 20 yards when Shelvey rolled a quick free-kick into him, but the Frenchman's involvement was curtailed when he pulled up with a hamstring injury after 17 minutes and was replaced by Marvin Emnes.
Federico Fernandez had also gone close after climbing well at a corner and Swansea thought they had a taken a 31st minute lead which would have been entered into the goal of the season category.
Shelvey teed himself up on the volley to flash a ferocious shot past Howard from 25 yards but the referee had spotted a foul on Baines by Wayne Routledge.
The close shave appeared to galvanise Everton and they finished the half stronger with Coleman's low drive forcing a wonderful fingertip save from Lukasz Fabianski.
Coleman was raiding down the right and Ross Barkley was drifting into dangerous pockets of space but when the goal came it was manufactured down the left as Baines and Arouna Kone combined to good effect.
James McCarthy raced onto Kone's pass and his cross evaded everyone for Lennon to score in consecutive away games for the first time in his Premier League career.
Substitute Steven Pienaar tested Fabianski right on half-time and Swansea manager Garry Monk responded by changing his formation from a diamond system to 4-2-3-1 with Gylfi Sigurdsson the player pushed furthest forward.
Yet Everton were still dictating the tempo of the game and Gareth Barry might have doubled their advantage from a Baines corner, but his header lacked power and Fabianski safely gathered.
Emnes tried to inject some urgency into Swansea's play with a direct run which ended with a low shot past Howard's upright and Ki Sung-yueng produced a delicious bit of skill to lift ball over Barry, although he was unable to keep his shot down.
Swansea's pressure was rewarded when Emnes ran at Coleman and the Republic of Ireland defender appeared to have the situation under control but as he fell, perhaps expecting a free-kick for a shove in the back, he bizarrely handled.
Shelvey stepped up after some period of time which saw Howard booked for time-wasting to drill the ball low into the corner.
Howard acted smartly to tip over Sigurdsson's 25-yard free-kick and Everton had a chance to pinch it at the end - but substitute Kevin Mirallas shot straight at Fabianski and honours were shared.
Source : PA
Source: PA