Harriers looked like they were still stuffed with turkey when Swansea breezed into the lead within the first 40 seconds.
Paul Connor was allowed the freedom of the penalty box to glance a header past Ryan Clarke after Kidderminster failed to deal with a nothing ball from Andy Gurney.
Harriers switched off again on nine minutes when out of sorts keeper Clarke flapped at Roberto Martinez's cross and the ball bounced in off Abdou Sall for an own goal.
The sending-off of Gurney for kicking out at Simon Russell on 29 minutes evened things up a little, but Kidderminster lacked the quality to get back into the game.
Gary Birch threw the Aggborough men a late lifeline when he headed in a left-wing cross from Jon Beswetherick to make it 2-1 on 82 minutes.
A late collapse from confidence shy Kidderminster sparked chants of, 'You're not very good' from the visiting Swansea fans.
Andy Robinson restored the Welshmen's two-goal advantage from the penalty spot after Liam Burns clumsily upended Marc Goodfellow on 88 minutes.
Goodfellow got on the scoresheet himself, with a clinical finish after being sent free by Kevin Nugent on 90 minutes.
Robinson completed the rout with his second and Swansea's fifth in stoppage time with a drilled shot from the edge of the box.