The Glassboys survived the odd defensive wobble amid long spells of control and never looked back after notching twice in five minutes midway through the first half.
Dudley duo Callum Gillies and Brad Mansell tried to test debutant keeper Tyrese Warmington, drafted in on loan from Birmingham City in the week, early on.
Stour took the lead with their first clear chance when Alex Moore smashed it inside the near post after Joel Kettle had held it up in the box.
Jessy Bavanganga headed a freekick back across goal for Ethan Freemantle to scramble home the second.
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