Substitute Dimi Papadoupolus celebrated his birthday a week late by sending Burnley into the third round of the Worthington Cup.
The striker turned 21 last week but had to wait until the 13th minute of extra time to grab the goal that sealed Burnley's passage into the next round for the first time in 20 years.
The Greek Under-21 captain collected Tony Grant's waited throughball and held his nerve to slot the ball calmly past goalkeeper Scott Bevan with a minimum of fuss.
But Huddersfield, without a goal in their previous four matches, were certainly not finished and would have earned themselves a late reprieve but for the goalkeeping heroics of Marlon Beresford.
Town centre-half Nat Brown crashed a 30-yard effort against the post with just four minutes left on the clock. But with the goal at his mercy Jon Stead's follow-up was somehow kept out by Beresford after he launched himself to smother the shot.
Still the home side poured forward but Danny Schofield could not believe his luck when Beresford again rescued Burnley with another close-range save in the dying seconds of the match.
Despite their recent poor form the Terriers should have been two goals up within the opening 11 minutes only for Stead and Schofield to both miss the target.
At the opposite end winger Martin Smith was back on his line to clear Gareth Taylor's header from a Burnley corner.
Both sides created and squandered chances in the run up to half time but neither could break the deadlock.
Burnley captain Steve Davis' powerful header was destined for the top corner in the 68th minute before Bevan pulled off a superb fingertip save.
But with 11 minutes left of normal time Huddersfield contrived to miss a wonderful chance given to them on a plate by defender Mark McGregor.
Stead seized upon his under-hit back-pass as Beresford pushed the ball away from his feet and Schofield's follow-up shot was cleared off the line by the recovering McGregor.
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