It was a close first half with Drom leading by 12 points to 9 at the interval.
They pulled clear with second half goals from Tommy Nolan, county senior Seamus Callanan and David Butler.
The result in Holycross means Drom & Inch progress to the next round of the Mid championship.
Club secretary Matty Ryan was pleased with the victory but feels Drom have more work to do.
