Both sides had early chances at the Madejski Stadium but neither side managed to break the deadlock.
Darius Charles scored the only goal of the game for Stevenage with a well-struck shot from 20 yards out in the 21st minute following an assist from Scott Laird.
Reading were rocked by the goal and Brian McDermott's side failed to recover as they trailed the League One side at half time.
A much-improved Reading appeared in the second half. Michael Antonio wasted their best chance as he sliced his shot wide of the Stevenage goal.
Stevenage almost doubled their lead late on but Adam Federeci denied Joel Byrom with a fine save.
The League One club survived late pressure and held on for a 1-0 win to book their place in the fourth round.