Clarets boss Sean Dyche admitted Crystal Palace were the better side at Turf Moor.
But he was pleased with the way his side defended to claim a first home win of the season and first clean sheet.
Burnley now have seven points from their first four games, and Dyche said: “Against West Brom, I thought it was arguably our best performance of the season and we got nothing, and today wasn't the case, we got the win but had to fight for every inch of it.
“I thought they were very good, they played with more freedom than in recent weeks, and maybe the goal strangely helped with that.
“When you're 1-0 down you just think ‘right, let's just have a go’.
“I thought they were very strong and looked a more familiar side on paper, which I was worried about.
“I saw the teamsheet and thought it looked more like how Palace are and how they're effective.
“And they played like that, they mixed it well, put a lot in the box, for the strength of Benteke, and got it forward cleverly.
“In balance, we defended fantastically well and you earn a bit of luck now and again.
“They had a few chances but two golden chances, especially big Dann’s header at the end.”
Nick Pope came on for skipper Tom Heaton for his Premier League debut, after Heaton suffered a shoulder injury.
And Pope made a big stop to deny Christian Benteke to preserve the win: “It was a brilliant save, not just because he got his foot there, but he shortened the angle to go at the ball, and he delivered a good performance. It's tough when you go into that game, we’re not playing as well, the opposition are, and you've got to hold your nerve and he did that very well - and the two centre halves in front of him, Tarky was immense.
“Ben Mee was solid again, and we know we've got to fight for everything continually, and we've deserved what we've got away from home, we didn't play as well today but got the three points, and that's important.”
Palace boss Frank de Boer went into the game with speculation that a defeat could lead to his demise, and Dyche had a word with him after the game: “I don't think any manager wants sympathy, I just said ‘I thought you were the better side’.
“I think I'm honest, I attempt to be, and they were the better side, they deserved something from the game.
“I think there's a respect among managers and try to call it right.
“In balance, we've done the jobs we needed to to see the game out, but it's fair to be honest, they generally were the better side.”