An own-goal from Adrian Whitbread put the visitors ahead, and the result was never in doubt.
The win puts the Harriers onto 49 points, one above United.
Bury vs Oxford and Macclesfield vs Leyton Orient were both victims of the weather, as was Carlisle United's LDV Vans Northern Final against Shrewsbury Town.
Cambridge United's hopes of reaching the Milleniu Stadium were hampered when they were beaten 4-2 at Bristol City.