Beckett, an infrequent starter who recently said he would leave the club if not picked more regularly, put Town ahead inside four minutes when City keeper Maik Taylor failed to hold Danny Schofield's close-range shot.
Ex-Oldham and Chesterfield star Beckett, in the side as first-choice frontman Andy Booth was injured, then set up Chris Brandon for a late winner after Garry O'Connor had levelled for Birmingham on 19 minutes.
The result was even more impressive as Huddersfield fielded a patchwork side featuring midfielder Michael Collins as a makeshift right-back and 19-year-old James Berrett, making his debut, against a Birmingham team with four internationals in their starting line-up.
It was an exciting end-to-end contest that made for an entertaining introduction to FA Cup football in England for new Blues boss Alex McLeish.
Huddersfield nearly went back in front soon after O'Connor's leveller when Franck Queudrue was booked for a foul on Malvin Kamara.
From the resulting free-kick, swung in by Robbie Williams, Brandon had a goalbound header blocked on the line by Taylor's legs.
Gary McSheffrey, a £4million buy from Coventry City under ex-Birmingham boss Steve Bruce, then took charge of the contest with a storming individual display after the break.
He nearly set up Mikael Forssell for a goal within seconds of the re-start with a throughball which left the ex-Chelsea striker with an easy chance that he wastefully blasted straight at Town keeper Matt Glennon.
McSheffrey had a couple of long-range digs after that before coming close to scoring a second equaliser for the visitors when Glennon was forced to kick away his goalbound shot from a tighter angle in stoppage time.
But Huddersfield clung on to reach the FA Cup fourth round for the first time since 1999. Birmingham are Huddersfield's first top-flight Cup victims since QPR were beaten some 23 years ago.