The visitors had turned in a disciplined defensive display in the opening 45 minutes but a Phil Bardsley own goal was swiftly followed by a superb Jonathan de Guzman finish and a Wilfried Bony spot-kick after the break.
Chico Flores' header was deflected by Steven Fletcher for the hosts' fourth to give Swansea their first win over Sunderland since 1982 and a first at home since March, leaving Poyet’s men rooted to the bottom of the Premier League.
The hosts were without captain Ashley Williams, who remains sidelined with an ankle injury - but Ben Davies recovered from a hamstring problem in time to make the starting line-up.
|By Tom Davis
"This was not the start Gus Poyet would have hoped for as new Sunderland manager.
"Actually, it was. Defensively, the Black Cats were sound in the first half and looked on course for a solid point away from home. It was just the ending that was so agonisingly awful for the manager to endure.
"Their organised first-half performance counted for nothing when three second-half goals in eight minutes - including an own goal, a penalty and a stunning shot from distance - wrapped up victory, which was Swansea's first home win in the league since March."
Poyet, meanwhile, brought Bardsley, Sebastian Larsson and Steven Fletcher – returning from a shoulder injury - in for Jack Colback, Jozy Altidore and loanee Ki Sung-Yueng, who was unavailable for selection against his parent club.
And Bardsley – making his first-team return after a controversial period in the wilderness under former manager Paolo Di Canio – made an early impression as he was booked for a reckless challenge on Nathan Dyer.
But Sunderland were soon making more positive moves as Fletcher fired a left-footed strike narrowly wide from an Adam Johnson corner within the opening 15 minutes.
With Swansea struggling to find their stride, the visitors continued to press as Larsson hooked a seemingly lost cause into the penalty area from a tight angle, only to see goalkeeper Michel Vorm beat Fletcher to the cross.
The hosts had to wait until just after the half-hour mark for a shot at goal but De Guzman sent his 25-yard free-kick beyond the right-hand post as an even first half ended goalless.
Bardsley then had a chance to mark his return with a goal shortly after the break but he scooped his effort over the crossbar from six yards after a cross from the right.
However, Swansea soon began to find the flow that had eluded them in the first half and, after decent chances for both Dyer and Bony, they broke the deadlock in the 55th minute as Bardsley diverted the ball into his own net from a corner - the Swans' 100th goal in the Premier League.
Laudrup's side then doubled their lead within a minute as De Guzman got a sight of goal from 20 yards and curled a wonderful effort into the top right-hand corner of the net.
And they were three up six minutes later. Craig Gardner brought down Britton just inside the penalty area, allowing Bony the chance to send the ball beyond Keiren Westwood from the spot.
Chico Flores completed the rout in the 80th minute as he met a corner to head past Westwood via a Fletcher deflection. A miserable opening day for Poyet was complete.