The visitors enjoyed their share of the early possession and the best chance of the first half fell to Steven Fletcher, whose half-volley just evaded the post.
Swansea came closest to breaking the deadlock through a Jonathan de Guzman free-kick, but his curling effort from distance was unable to find the top corner.
After an evenly-matched start to the second half, the Swans eventually took the lead on 57 minutes when De Guzman's corner was inadvertently turned into his own net by Phil Bardsley.
Moments later Michael Laudrup's side doubled their advantage with a fine strike from De Guzman as the Dutchman collected the ball 25 yards out before curling past a helpless Keiren Westwood.
The Swans were then awarded the chance to make it three and put the game beyond Sunderland's reach when Leon Britton was tripped by Craig Gardner inside the area. Wilfried Bony calmly stepped up and slotted home for his first goal since the opening day (64).
Sunderland's misery was compounded 10 minutes from time when Chico Flores stooped to meet De Guzman's corner, with his header diverting off the legs of Fletcher and into the net.
Due to the international break, Poyet had only had his full squad available to him for a couple of days of meaningful preparation, but there had been some positive signs for the Black Cats during the early going.
But the manner in which they crumbled, conceding the first three goals in just seven second-half minutes, highlighted that Paolo Di Canio is not the sole reason Sunderland prop up the league table.
At times, they had looked relatively comfortable moving the ball around in a similar fashion to their manager's teams at Brighton, but they were helped no end by a sloppy Swansea start.
Chico had to make a crucial tackle to prevent Adam Johnson racing clear after a terrible pass by De Guzman, while Fletcher should have had the Black Cats ahead a few minutes later.
Swansea were caught sleeping at a corner as Johnson found Fletcher unmarked, but the Scotland forward could not control his half-volley and his shot flew wide.
A De Guzman free-kick flew just wide of Westwood's post, but it was as close as Swansea came as Sunderland dropped men deep around their 18-yard box to frustrate their hosts.
Bardsley should have scored at the correct end early in the second half, but slipped after Emanuele Giaccherini's ball had run all the way across the box.
But Swansea, with some harsh words from their manager ringing in their ears, were soon creating clear chances at regular intervals.
Firstly Michu blazed a volley over after good work by Bony, before the Ivorian failed to convert Angel Rangel's low cross from six yards.
The opener duly arrived in the 57th minute, although there was a healthy dose of luck involved.
Rangel flicked on De Guzman's corner where it hit the unfortunate Bardsley to beat Westwood.
The Black Cats had no time to settle before the deficit doubled, De Guzman curling home a beautiful effort from the edge of the area to rival his stunning strike in Valencia last month.
Westwood then made a sharp stop to deny the Holland international's volley, but it was 3-0 in the 64th minute as Gardner brought down Britton to give Swansea a first Premier League penalty in 56 games.
Bony made no mistake from the spot for his sixth Swansea goal, before Chico bundled in a header off the legs of Fletcher from another superb De Guzman set-piece 10 minutes from time to give the Welsh club their biggest home win in the Premier League and leave Poyet with plenty to ponder.