Huddersfield were nearly rewarded for a superb start when James Vaughan found space inside the box but the striker couldn't hit the target with his low effort.
Lee Clark's side failed to improve following their sluggish beginning to the match but the visitors were guilty of wasting another chance when Peter Clarke headed over from close range.
The Terriers finally had the lead that their play deserved after Vaughan broke free of the Birmingham defenders before picking out Beckford and the on-loan forward calmly finished from six yards out.
Nathan Redmond went close to equalising before the break but he could only hit the post with a curled shot from the edge of the area.
Birmingham appeared to be a different side after the break, with Alex Smithies being forced to deny Chris Burke just minutes into the second half.
Huddersfield once again managed to take control of the match and a second goal for the visitors looked more likely for large spells of the second half.
Clark had introduced Nikola Zigic at half-time to put pressure on the Huddersfield defence but the striker's afternoon ended earlier than expected when he was shown a red card for a dangerous tackle on Calum Woods.
After the dismissal Birmingham looked like snatching a draw but some stubborn defending secured the result for Huddersfield.