The visitors took an early lead through midfielder Aaron Ramsey and were forced to hold on for the remaining 78 minutes.
The victory extends the Gunners' winning streak to eight in all competitions and leaves the Londoners just four points behind Premier League leaders Chelsea.
And Wenger was pleased with how his players dealt with an organised Burnley side.
We did enough to deserve to win the game
Arsene Wenger
"It was a battling performance more than a fluid performance, and that was because Burnley are perfectly organised and very fit to keep their shape for the full 90 minutes.
"They made life very difficult for us. The first goal was vital and we were a little bit in between: do we go for another or try to keep it tight.
"That is not always the best place to be. When you have not scored the second goal you are always under threat, but I thought we did enough to deserve to win the game."