A strike from the spot earned a Premier League point for the Londoners at the Liberty Stadium last season, and history repeated itself this time around.
Wilfried Bony gave the home side an excellent start inside quarter of an hour, turning his man to fire home, before Jedinak equalised with a powerful spot-kick for his fifth goal of the campaign, after Jonjo Shelvey had brought down Marouane Chamakh.
Shelvey nearly made amends for his error after the break, only to volley straight at goalkeeper Julian Speroni.
However, a spirited second-half performance from Palace ensured they held on for a point, although Neil Warnock's men have now gone four away games without a win.
Despite a setback at Manchester City last week, Swansea came into the game having made their best Premier League start, largely due to an impressive home record that had seen them beaten just once in the league all season.
Boosted by the return of attacking duo Shelvey and former Palace winger Wayne Routledge, the hosts soon set about asserting themselves once again during the opening five minutes, with Bony heading just wide after some good work from Jefferson Montero.
Gylfi Sigurdsson flashed a free-kick just over the bar three minutes later, as the visitors struggled to find the fluency that saw them triumph over Liverpool last weekend, despite having only made one change to that winning side.
Palace had previously never won a Premier League game at the Liberty Stadium, and Warnock's side were behind inside 15 minutes when Sigurdsson picked out Bony in the area, with the Ivorian shaking off the attentions of Brede Hangeland to drive home for his sixth of the season.
The visitors were dealt a further blow when Dwight Gayle was forced to limp off, but they were given an unlikely route back into the game after 25 minutes when Shelvey was adjudged to have tripped Chamakh inside the box, with Jedinak duly converting from the spot.
That goal galvanised Palace, with Chamakh forcing a superb save from goalkeeper Lukasz Fabianski four minutes later, before Yannick Bolasie saw his vicious strike from distance tipped over the bar on the half-hour mark.
The pace of the game began to slow after the break, although Shelvey should have put his side ahead on 52 minutes, volleying straight at Speroni from close range.
Speroni had to alert 10 minutes later, though, producing a smart save to deny Sigurdsson from distance, in what was otherwise a reasonably quiet second half.
The shortage of chances saw Swansea bring on Bafetimbi Gomis 18 minutes from time as boss Garry Monk looked for a winner, which nearly came four minutes later when the industrious Sigurdsson fired a low drive just wide from outside the box.
Swansea continued to press, but Palace held firm to deny the hosts a third consecutive home win.
