Mark Sheeran, Darlington's teenage striker, jumped off the bench in the dying minutes of the match to score two late goals and bag a 3-2 victory for the Quakers.
Luton took the lead after 13 minutes when Steven Howard headed in.
The home side levelled with 28 minutes gone when defender Gary Pearson rifled in a 25-yard free-kick.
But Luton went back in front when French midfielder Jean Luis Valois launched a curling left-foot drive on the hour.
Sheeran then grabbed his first with nine minutes remaining, firing in after goalkeeper Mark Ovendale had parried Richard Hodgson's shot.
He added his second three minutes from time as he headed Pearson's cross over the stranded Ovendale.
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