Scott Sinclair's eighth-minute strike was enough for the Swans to take the points in a game where both sides ended with 10 men following the dismissals of Darren Pratley and Tyrone Mears.
The Swans have won all three games on their own patch this season following wins over Preston and Barnet, and Rodgers is hoping to gain maximum points from their home games this season.
"That is four wins out of six this season and we want to make the Liberty Stadium a real fortress. That's two Championship wins in a row here and we've scored five goals - that's a good start," he said.
"We've got two players out wide with real pace who can cause problems for the opposition and the idea is to get them playing as high up the pitch as possible.
"I like my teams to be offensive and creative and I think the fans enjoyed their afternoon. We were pretty good value in the first half.
"It was a real test for us against a team that had just come down from the Premier League and I thought we passed the test really well.
"It was a great statement that people can identify us as a good footballing side that can also show a bit of steel and character.
"I think it could be exciting over the next two or three seasons. We have played well in every game and I can only see us getting better.
"I want to build a squad that can be really, really competitive in a league that has a lot of money in it."
Burnley manager Brian Laws has slammed Mears for his red card, which prevented the Clarets from pushing for an equaliser.
Mears walked less than a minute after Pratley had seen red for the Swans when he picked up his second booking.
"That game shows that Swansea City are a good team. We knew that before we arrived and we didn't take them lightly," said Laws.
"Yet the frustration is that we never had the opportunity to taste the extra man. We had about 30 seconds of 11 v 10. We would quite clearly have gone on and won the game had we had longer.
"Tyrone Mears has quite clearly got to look at himself. We are professionals and we have to play with our heads and not our hearts. Even the most experienced players lose their heads sometimes."