Aaron Ramsey's 12th-minute strike made it eight successive wins for Wenger's men - the most prolific run by any team in the English top flight this season - but they were made to work for their victory.
The result leaves Burnley second bottom and two points from safety with six games remaining, but Wenger hopes they are able to beat the drop.
"I'm pleased with the performance because it was more a fighting performance than a fluent performance and many people question us on that side," he told his pre-match press conference.
"I must say that I'm surprised by the quality of Burnley.
"It would be a shame if they go down becuase they are fantastically well organised, they press very well, they have a great solidarity and they are very fit.
"I understand now why they took points from the big teams here.
"We had to fight; we had to be combative until the end - we couldn't score the second goal.
"Our defenders played well today and that got us the three points. Our defenders, keeper included, deserve credit."
Arsenal's stellar form and the collapse of reigning champions Manchester City over recent weeks has established Wenger's team as the closest challengers to league leaders Chelsea.
The Stamford Bridge outfit remain four points clear with two games in hand, however, and Wenger is understandably reserved over the prospect of a late Arsenal title charge.
"Nothing has changed," he added. "They are still too comfortable, Chelsea.
"We need a perfect run and we need a non-perfect run from them.
"So we cannot master that. We can only master our own attitude. I feel they are still in a comfortable position."
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