The Premier League leaders eventually triumphed 2-0 through goals from Eduardo and Nicklas Bendtner but Burnley, mid-table in the Championship, nearly took the lead when Kyle Lafferty hit the crossbar.
Lafferty was later sent off by referee Alan Wiley for a studs-up challenge on Gilberto, after which Bendtner's goal ended the contest.
Wenger said: "Burnley played very well. They missed the first big chance and I believe that header could have had a big weight on the game.
"With the performance they put in I think it would have been a very difficult game for us if they had gone 1-0 up.
"We then took our chance, but Eduardo missed another good chance to score a second - the surprise of the day is that he missed it - you quickly get used to him scoring a goal when you want him to but he scored one and got an assist for the second so overall I believe the two strikers did well.
"Credit to Burnley, they played very well and in a fair way and with the way they play I am surprised they are in their position in the Championship - perhaps it's better to be in the Premier League!"