The goal came when Gary Teale's 52nd-minute corner was met by Buxton at the far post with a firm downward header.
Leyton Orient will feel they should have secured at least a point after creating a number of good openings.
But they were left to rue a combination of wasteful finishing and an impressive performance from Wednesday keeper Nicky Weaver.
When Weaver was beaten, Darren Purse and James O'Connor came to his rescue, clearing the ball off the line from Dean Cox and Scott McGleish respectively.
The busy Cox also fired off target from a good position inside the area and forced a good save from Weaver.
Alex Revell was the first to test the Owls stopper with an early header while Terrell Forbes headed wide later in the half.
Aside from the goal, Wednesday's only other effort was an angled shot from Teale which was parried by Lee Butcher.