The diminutive winger had netted three Premier League goals in five games coming into this clash at the Liberty Stadium, and he made it four for the campaign with a composed strike in the first half.
Gylfi Sigurdsson then added a second after the interval as he reacted first to Sylvain Distin's wayward header coming back off the crossbar before substitute Marvin Emnes fired in a third with three minutes remaining.
Roberto Martinez, a former player and manager at Swansea, was unable to inspire his Everton side to a second-half comeback as they fell to back-to-back defeats following Sunday's surprise 3-2 home reverse at the hands of Crystal Palace.
For the visitors, a third game in six days was perhaps just one too many after their UEFA Europa League exploits last Thursday.
Meanwhile for Swansea, winners of this competition under Michael Laudrup in 2013, victory was a welcome boost after successive defeats to Chelsea and Southampton in the Premier League as they secured their first ever win over Everton after 20 previous failed attempts.
Swansea manager Garry Monk made five changes from the 1-0 reverse to Southampton at the weekend, while Everton’s side included seven alterations from their loss to Palace.
Bryan Oviedo made his first appearance since breaking his leg in January, and there were full debuts for Luke Garbutt and Muhamed Besic.
Sigurdsson twice fired off-target in the opening four minutes, before Samuel Eto'o shot wide with Garbutt better placed to his left.
Antolin Alcaraz, making his first start of the campaign for the visitors, was denied at close range by Gerhard Tremmel after the hosts failed to clear Garbutt's free-kick from the left in the 18th minute.
Dyer put the hosts in front 10 minutes later with a low finish at the far post after excellent work down the left by Jefferson Montero.
Martinez introduced Romelu Lukaku off the bench to replace Eto'o at half-time, and the Belgium international lashed a shot over the bar within 40 seconds of the restart.
Aiden McGeady did likewise five minutes later, while Tim Howard palmed away Sigurdsson's header after yet more incisive play down the left flank by Montero.
The Iceland international was not to be denied in the 64th minute, however, as he stabbed home from close range after Distin had diverted Jonjo Shelvey's cross onto the woodwork.
Tremmel had to race from his goal to deny Lukaku 11 minutes from time, but it was Swansea who pulled further clear in the final stages as Emnes' low drive from 20 yards put the seal on their place in the fourth round.
