Three goals in six second-half minutes from Luton's Ahmet Brkovic turned this game on its head and ensured the Hatters made progress in the LDV Vans Trophy.
Stevenage had made the early running and were good value for their 2-1 half-time lead but they were to rue two first-half missed chances as Brkovic ran riot after the interval.
Stevenage did battle back and were handed a lifeline 19 minutes from the end when Richard Scott curled in a spectacular free-kick.
Stevenage weren't able to capatilize and were denied a leveller in the dying seconds when Carl Emberson pulled off a stunning save from Gary MacDonald's close-range effort