Jemale McKenzie-Lowe latched on to a through ball from Charlie Smith before rounding the keeper to score.
Luton’s advantage didn’t last long though as Stevenage hit back twice to move 2-1 in front.
As the game entered the final five minutes, Town levelled when Burtenshaw found Smith whose cross was put through his own net by a Stevenage defender.
Ben Barnes denied Boro from regaining the lead straightaway, before Town sealed the points in injury time when Leeroy Maguaraushe found Amu and he neatly placed the ball past the keeper for a dramatic win.
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Hatters: Barnes, Burtenshaw, Galliford, Nash, Jibodu (C), Chabata, Cobb (Davis 59), Amu, McKenzie-Lowe, Maguraushe, Smith. Substitutes not used: Carpenter, Stanley, Rees.
For a full report and the thoughts of U18s boss Paul Driver, see the Luton News/Dunstable Gazette on sale tomorrow, while for all the latest Luton Town FC news, reports, stats, player profiles, live match updates, MOM vote, prediction table and a complete fixture list, visit our Hatters Microsite.