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 the Swans, who triumphed 2-1 at Manchester United in their first match, 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."
Burnley boss Sean Dyche was frustrated by the result but felt that his team had shown evidence of further acclimatisation to the top flight after being beaten by Chelsea
"It's disappointing that we didn't come away with anything, but you could see the growth from us," he argued. "In the second half, we were excellent.
"Sometimes, when you are promoted, you are wondering if you belong here. In the first half, we showed them too much respect and at half-time I said: 'Go out and express yourselves'.
"The movement, energy and determination was good but we have to learn quickly. You have to remember what you are and believe in what you are.
"We knew this would be a challenge, a difficult one, and we are enjoying the way the team are operating but we need to get points on the board."