Michu and Nathan Dyer scored early in either half to hand Michael Laudrup's side the points and inflict what was only Palace's third defeat in 27 matches in all competitions at Selhurst Park to leave them second bottom of the table.
Home manager Ian Holloway, serving the last of his two-game touchline ban, made three changes from last week's defeat at Manchester United with Barry Bannan coming in for the suspended Kagisho Dikgacoi for his debut, while Jose Campana and Dwight Gayle made way for Adlene Guedioura and Cameron Jerome.
Laudrup, meanwhile, made five alterations to the Swansea side that was so impressive in winning 3-0 at Valencia in the UEFA Europa League on Thursday.
Ashley Williams, Ben Davies, Jonjo Shelvey, Wayne Routledge and debutant Alvaro Vazquez all came in to replace Jordi Amat, Dwight Tiendalli, Jonathan De Guzman, Alejandro Pozuelo and Wilfried Bony.
But it was Michu who gave the visitors the perfect start, as Shelvey's pass ricocheted kindly off Routledge into his path before the Spaniard fired under the legs of Julian Speroni after just two minutes.
And it was almost two a minute later, as Chico Flores met Shelvey's corner at the back post only to see his effort cannon back off the woodwork.
Laudrup's side were well in control, and Michu should have doubled his tally midway through the half but he inexplicably decided to attempt a backheel towards Vazquez with the goal at his mercy.
The Spaniard then went close again on the half-hour mark, Danny Gabbidon stretching out a leg to deflect his goalbound volley just over the crossbar.
It took 35 minutes for the hosts to muster a meaningful effort on goal - Guediora's strike from 30 yards flying straight into the arms of Michel Vorm - before Angel Rangel almost scored a spectacular second for Swansea when his sweetly struck volley was beaten behind by Speroni.
The visitors picked up where they left off after the break, and were rewarded for their dominance when they doubled their advantage almost immediately.
Vazquez saw his shot parried by Speroni, but picked up the loose ball on the byline and cut it back for the unmarked Dyer to sweep home from 10 yards.
Vazquez then wasted a golden opportunity to score on his Swansea bow in the 56th minute, firing over with just Speroni to beat following good interplay between Michu, Dyer and Shelvey.
Palace nearly got themselves back into the game with 20 minutes left to play with their first effort of note in the second period, but substitute Gayle's effort from outside the area was deflected narrowly wide.
But Holloway's men got little joy out of the Swansea defence in the closing stages as the visitors inflicted their fourth defeat in five Premier League outings this season.
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