The Norwegian curled home after Ashley Westwood had clipped Bukayo Saka on the edge of the area.
It was Odegaard’s first goal since joining the Gunners on a permanent basis from Real Madrid in a £30m deal last month.
Burnley were denied a chance to equalise from the penalty spot when referee Anthony Taylor overturned his initial decision after reviewing Aaron Ramsdale’s challenge on substitute Matej Vydra on the pitchside screen.
It was the high point of a spirited second-half performance by the hosts, whose manager Sean Dyche was this week given a new four-year contract and handed a debut to Ivory Coast wide-man Maxwel Cornet.
A second successive win eases Arsenal into mid-table following their poor start to the campaign, but Burnley remain second from bottom with a single point to show from their opening five games.
As they had not lost at Turf Moor since 1973, Arsenal travelled north with some degree of confidence.
They were helped in extending their positive record by a strangely subdued home side, who were not a real threat until Cornet’s introduction after 57 minutes.
Arsenal’s midfield axis of Odegaard and Emile Smith Rowe either side of a deep-lying Thomas Partey was particularly effective, even if the goal threat of Nicolas Pepe and skipper Pierre-Emerick Aubameyang was negligible.
Still only 22, Odegaard still has plenty of time to prove why Real Madrid bought him as a 16-year-old in January 2015.
Although he only made 11 appearances for the Spanish giants, the Norwegian evidently has plenty of confidence.
He was happy to take responsibility from the set-piece after Westwood had fouled Saka and provided the perfect finish, over the ball and beyond the dive of Burnley goalkeeper Nick Pope.
Manager Mikel Arteta might have wanted more goal threat from his team, who dominated possession but surprisingly had fewer shots and corners than their hosts.
However, as the familiar “One-nil to the Arsenal” chant rang out at the end, Arteta could at least reflect on his side stabilising their terrible start to the season.