The Swans had not won on home soil in eight Barclays Premier League games and were frustrated by the Black Cats during a goalless first half.
But Phil Bardsley's own goal, a wonderful Jonathan de Guzman strike and Wilfried Bony's penalty - Swansea's first in 56 league games - completely altered the course of the match in just seven minutes.
Sunderland had been combative until the 57th-minute opener, with Fletcher and Bardsley having their best chances, but they 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.
Sunderland remain rooted to the foot of the table with one point from eight games, and Poyet is fully aware of the challenge ahead.
The Uruguayan said: "I learnt a lot from my players, particularly in the second half.
"My team talk at half-time was all about the ball and what we needed to do better with it to go and create something more.
"The rest of the game I was very pleased with as we had worked on being well organised and difficult to beat. Then we got to the second half and we did completely the opposite. It's difficult to explain.
"We have no other option than to work and turn that (first-half) performance into 90 minutes as soon as possible, as time and games will pass by with us still at the bottom.
"Any team can concede a goal, if you cannot play any more because you concede a goal, you deserve to be at the bottom."
Poyet also suggested he is prepared to make changes for next weekend's derby meeting with Newcastle having been unimpressed with how his men reacted to falling behind.
"I thought I picked the right team in the first half, now if you were to ask me if I did, I would not be sure," he said.
"They have given me a chance. As a player you do not want to give the manager the chance to pick someone else. They have given me the chance to do that whatever they like this week.
"What a game we have next week. It is becoming serious, it is a derby, we are at home, we are not winning games and now is the time to start."
Swansea had only been above the relegation zone on goal difference heading into the weekend, and their tension at the lack of a home league win since March was clear to see during an error-strewn first half.
But boss Michael Laudrup was delighted by how his side had turned it around after the break.
Laudrup said: "In the Premier League it has been a while since we won at home. We have won all three of our home games in the Europa League, but this is our first of the league season.
"We can always talk about who we have played against, and we have played three of the top teams here this season. But when people start to talk it is good to turn it down immediately, and the only way to do that is by winning."
The only negative for Swansea was an ankle injury suffered by left-back Ben Davies, who was withdrawn early in the second half and left the Liberty Stadium wearing a protective boot.
Laudrup confirmed the Wales international will undergo an X-ray to determine the severity of the damage.
He said: "It's another ankle injury, we have already had a few.
"It is a twisted ankle, he fell in the first half so we agreed to let him start the second half but after two or three minutes you could see it wasn't good.
"He will probably go for an X-ray, but I am not a doctor and we have to wait for the X-ray. But I think he may be out for the next two games (against Kuban Krasnodar and West Ham) at least."