Swans' boss Gary Monk, embarking on his first full season in charge, saw a goal from our former loan player Nathan Dyer win the points for his side before they had their work cut out in the second half to keep out the Clarets.
"It was a very hard fought three points," Monk admitted after the game. "We talked at half time about how Burnley would come and throw everything at us. In the first half, I felt we pretty much controlled the game and should have got a second, but when we weren't at our fluid best in the second period, we showed a different side to us and dug in.
"There are times when you have to dig in, and I felt the boys were magnificent in that respect. We defended very, very well and limited them to one shot on target the whole game. A lot of that was down to the work we have done in pre-season, which was nice to see.
"It would've been nice to control the game better in the second half but, all in all, we did well."
Monk added, on their start to the season: "We can do better. It's great to have six points, but the players know that you are only ever one or two games from people being on your back. We have a game on Tuesday, which is what we'll focus on now.
"We will try and improve on the mistakes we make and, ultimately, we want to try and win as many games as possible."
Sean Dyche saw his side go behind to that first half goal before giving that second half scare, and he saw plenty to please him.
"We are a work in progress and we are developing," he said. "I think we have had a good week in the sense of the challenge that's been placed upon us, but we have to get points on the board and we have to win games, and we will.
"We have to be encouraged because we've been written off by everyone. We are the only ones who can encourage each other, along with our fans who were fantastic again today.
"But, as I suggest, I am someone who likes winning and we didn't get that today, so the challenge is to turn those good performances into results."
He added: "It's a cruel game sometimes, particularly the higher the level you go. The Premier League is unforgiving, but I felt the quality of our play earned us something. That's not always the way it works, but I thought we were outstanding in the second half.
"In the first half their quality of play was better than ours and I thought we showed them too much respect if I am honest, but it's only a belief thing. The only difference is their players have been here a bit longer and they believe they are Premier League players.
"Ours are still growing in that belief and that was evident today. They are finding their feet in this division and some are more assured than others, just by the way footballers are, but in the second half we had the movement and the passing and we created enough to get something out of the game, for sure. Sometimes it just doesn't go your way.
"Last year, people who saw us know we are relentless in the way we go about our business. We don't worry if we go a goal down and I want the players to continue to play on the front foot and go after games. It was a really poor goal from our point of view at a time when they had the upper hand.
"That freedom I talk about was on show in the second half, but we just didn't have the cutting edge to make use of it, alongside a fantastic save from Fabianski."