Nathan Dyer scored the only goal of the match in the 23rd minute and the home team looked good value for their lead for the remainder of the first half at the Liberty Stadium.
However, Burnley came out fighting after the break and Swansea, who triumphed 2-1 at Manchester United last weekend, had to be on their guard to maintain their winning start to the season.
"I knew it would be a tough game," Monk told BBC Sport.
"In the first half, we controlled it, but in the second we expected them to throw everything at us and they did. But you have to applaud our players.
"You have games like that and it's about the attitude of your players and the way they defended was superb, restricting Burnley to one shot on target.
"We matched them toe-for-toe and we defended properly.
"We can improve in many areas, as we didn't pass the ball as well as we can do in the second half, but our attitude and defending was great."
Swansea meet West Brom at the Liberty Stadium next weekend, but must first contend with a League Cup clash against Championship side Rotherham United on Tuesday.
