Second-half goals from Lucas Akins, Luke Freeman and Filipe Morais saw Stevenage end Stourbridge's cup run with a 4-0 win.
Akins netted four minutes after the break while Freeman added a third on the hour after Francois Zoko had broken the deadlock for Boro just before half-time.
Stourbridge, the lowest ranked team left in the FA Cup, never recovered after going behind and Morais completed the easy win from the spot at the death.
For 40 minutes Stourbridge were equal to the League One outfit but that all changed when midfielder Freeman's cross from the right was headed home by Zoko.
Akins doubled Stevenage's tally in the 49th minute, tapping in after his own looping header from a Dean Parrett cross had bounced back off the bar.
He then turned provider for Freeman, pulling back from the right 11 minutes later while Morais added a fourth in the 90th minute from the spot after he was bundled over in the box.