The Swans had not won on home soil in eight games but a Phil Bardsley own goal, a wonderful Jonathan de Guzman strike, Wilfried Bony's penalty and Chico Flores' header gave them a handsome victory.
Sunderland had been combative up until the opener, with Steven Fletcher and Bardsley having their best chances, but fell apart during the final half-hour to illustrate the task facing Poyet as he seeks to draw the Wearsiders clear of the relegation mire.
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 mananger's teams at Brighton, but they were helped no end by a sloppy Swansea start.
There were any number of misplaced passes from the hosts, enough to get Michael Laudrup up and out of his technical area on several occasions.
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.
Swansea's lethargy was summed up when Ben Davies, normally so reliable at left-back, allowed Sebastian Larsson too much room to get in a cross that Michel Vorm had to gather at the second attempt.
A De Guzman free-kick flew just wide of Keiren 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 strike 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 Craig Gardner brought down Leon 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 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 give Poyet plenty to ponder.
Source: PA
Source: PA