The Gunners, who have now lost four straight home games for the first time since 1959, are in 15th place in the standings with 13 points while Burnley climb out of the bottom three moving up to 17th place on nine points after their first away win of the season.
- Arsenal ratings: Xhaka 3/10 for reckless gesture in loss
The pressure will now increase on Spanish coach Mikel Arteta with Arsenal having made their worst start to a season in 39 years with their first home defeat by Burnley since 1974.
Mikel Arteta's side were enjoying a spell of strong pressure after the interval, testing Burnley goalkeeper Nick Pope, when Xhaka was sent off after a VAR review spotted him grabbing Burnley's Ashley Westwood by the throat.
The London side have received six Premier League red cards since Arteta took over Arsenal in December 2019, double that of any other side in that time.
Westwood created the decisive goal in the 73rd minute when his dangerous in-swinging corner was headed into his own goal by Aubameyang as he rose to challenge Josh Brownhill at the near post.
Arsenal created plenty of pressure in the latter stages but Sean Dyche's team held firm helped by the excellent goalkeeping of Nick Pope.
There were 2,000 supporters at the Emirates Stadium but some left before the end while one fan was pictured with his face mask pulled up over his eyes.
Arteta had no doubts that the red card turned the game against his side.
"We threw it away, we were in control of the game. When you give the red card away the game changes. I don't know what it is we have to do to score a goal," he said, before criticising Xhaka.
"We know that these things are unacceptable and the moment that we are in now it's even more so," he said.
Burnley manager Dyche was delighted his team finally got their first win against Arsenal in the Premier League era.
"I think the win gives us more belief. The fitness is getting better, the Premier League sharpness, the win adds to the belief but the next game comes around quickly," Dyche said.
"People have been reminding me that we never beat Arsenal, so that's another one down."