The Gunners scored an impressive home win over north London rivals Tottenham in their previous Premier League fixture, but dropping points at Turf Moor would have taken the sheen off that derby day success.
They looked to be settling for a point on Sunday but banked all three points when Alexis Sanchez converted a disputed injury-time penalty.
"For us, after the win against Tottenham, it was very important to go away from home, play a difficult game and win
That's what we did," said Wenger, whose side host Huddersfield on Wednesday.
"For the belief of the team it was a very important win.
"We kept a clean sheet
I congratulate our three centre-backs again because they did a remarkable job."
Teams who struggle at Burnley, and there have been many in the past two seasons of Premier League action, often fall back on stereotypes about their style of play.
References to long balls, second balls and physical football are common place from visiting managers but Wenger offered a respectful assessment of the division's surprise package.
He said: "They were really up for it, Burnley
For sure they did very well, they were very well organised, very clever, very intelligent, very direct as well.
"You understand when you play them why they have 22 points."
The Sanchez penalty - accepted, albeit through gritted teeth, by manager Sean Dyche - was the first goal conceded by Burnley in 375 minutes.
The only sides to have conceded fewer than their 10 goals so far are Manchester City and Manchester United, who sit first and second in the table.
Michael Keane, Tom Heaton and Jack Cork have all been capped for England while playing at Burnley and the presence of Three Lions boss Gareth Southgate gave Clarets captain Ben Mee hope of joining that roll call at some stage.
"Hopefully he has taken some positives from ourselves and a few of our players are fighting to catch his eye," he told the club's official website."I just want to play well for this club and do well and keep keeping clean sheets like we have done in the past few weeks.
"We were unfortunate not to do that again, so I just want to play well and whatever comes along with it comes."
Source : PA
Source: PA