The Gunners forced Nick Pope into a number of saves, but their hopes of sealing a first Premier League win since the victory over Manchester United on 1 November were hit when Xhaka was dismissed shortly before the hour mark for grabbing Ashley Westwood by the throat.
Their misery was compounded in the 73rd minute when Pierre-Emerick Aubameyang headed into his own net from Ashley Westwood's corner, to the dismay of 2,000 returning Gunners supporters.
Arsenal, which has now lost four consecutive top-flight home games for the first time since December 1959, remains in 15th with 13 points, while the Clarets move up to 17th place.
The host made 10 changes from the side that beat Dundalk in the Europa League on Friday (AEDT), and started slowly. It should have fallen behind after 13 minutes, but an unmarked Chris Wood inexplicably headed wide from eight yards.
Rob Holding headed over at the other end, while Alexandre Lacazette was denied by the leg of Pope after being picked out inside the penalty area by Kieran Tierney.
The Burnley goalkeeper denied Tierney and Bukayo Saka in quick succession at the start of the second period before Xhaka's moment of madness.
The Switzerland international's initial caution for pawing at Westwood was upgraded to a red card by referee Graham Scott after he viewed the incident on a pitchside monitor.
That was the sixth Premier League red card for the Gunners since Arteta took over in December 2019 – double that of any other side in the same period.
Mohamed Elneny was fortunate not to also receive his marching orders soon after, Scott deciding his push on James Tarkowski warranted just a yellow card.
Arsenal's reprieve did not last long, though, Aubameyang heading past Bernd Leno from Westwood's corner to guarantee an uncomfortable few days for Arteta.