"I knew it would be a tough game. 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," Monk told BBC Sport.
"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 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 return to action on Wednesday when they host Rotherham United in the second round of the League Cup.