A strike at the beginning of each half ensures the Swans carry on their impressive form following their fine Europa League victory over Valencia on Thursday.
The Swans made a flying start and were ahead after just two minutes when Michu blasted home, while Dyer swept in a loose ball three minutes after the break to double the winning margin.
Palace sits second bottom of the table following its third defeat in 27 home matches.
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 Guediora 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 débutante Alvaro Vazquez all came in to replace Jordi Amat, Dwight Tiandelli, 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.
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 was well in control, and Michu should have doubled his tally midway through the half but he inexplicably decided to attempt a back heel 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 goal-bound volley just over the crossbar.
It took 35 minutes for the home side 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 visiting side picked up where it left off after the break and was rewarded for its dominance when it doubled the 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 back into the game with 20 minutes left to play with, mustering its first effort of note in the second period, but substitute Gayle's shot from outside the area was deflected narrowly wide.
Holloway's men got little joy out of the Swansea defense in the closing stages as the visiting side inflicted a fourth defeat in five Premier League outings this season.