The visitors made a great start to the match and had created the best chance prior to Dyer's strike.
That opportunity fell to France striker Loic Remy - on a run of eight goals in nine league games - but his header was brilliantly saved low to his left by Michel Vorm.
Swansea have not lost this season when scoring first - picking up five wins and a draw
Swansea, unlike the Magpies, were without their top scorers, Wilfried Bony and Michu, who are still injured.
And Laudrup was left with fewer options when Alvaro Vasquez limped off with what appeared to be a groin injury.
The Spaniard, on loan from Getafe, had earlier gone close with a powerful angled drive that missed the Newcastle upright by no more than a foot. His replacement Dyer fared much better just before half-time.
Alejandro Pozuelo, who was also one of Swansea best performers, delivered a cross from the left that was tipped over by Krul to the corner of the six-yard area. Positioned there was the former Southampton player, who steadied himself before volleying into the Magpies net.
Swansea went close again shortly after the break when the energetic Shelvey launched a low drive that flew just inches wide of the Newcastle post.
They eventually doubled their lead - and the England international was heavily involved.
This time he picked up the ball and raced into the area only to see his shot parried by Krul. However, the ball struck covering defender Debuchy and bounced into the empty net.
Pardew unhappy with "blatant" penalty
A brave headed clearance by Newcastle substitute Massadio Haidara prevented Chico Flores from heading in a third, but the Swans were not to be denied moments later.
And appropriately, the final word went to Shelvey, who capped off a superb night for the Swans fans with a wonderful strike from outside the area that Krul could not keep out.
As for Newcastle, they barely threatened in the second half. However, they will look back at an incident with the scores at 1-0 when Remy's goalbound strike struck Ben Davies' arm - appeals followed, but they were dismissed by referee Howard Webb.
Newcastle boss Alan Pardew on the Davies handball:
"I'm still waiting for an explanation.
"The ball was going in the net and he [Davies] stuck an arm out. All our players have reacted. From the second they went up and scored. I'm aggrieved about the decisions.
"In all honesty though Swansea played terrific and we weren't good enough to cope at times. In the second half we pushed and when you push you need decisions to got your way - we've had two decisions go against us."
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