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.
Odegaard later said: "When we got the free-kick I immediately went to the ball. It was a big moment and a nice moment to see it go in. You need to hit it pretty perfectly, with the right speed and the right height.
"The game was kind of a war in the end. We know their strength. We played their game a little more than our own game. In the second half they played their game and they're good at it. We need to be a little bit smarter when we win the ball.
"I believe in this team. Even though we lost some games in the beginning, I'm not too surprised we've got some points now."
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