Carvalhal has made an impact since arriving at the Liberty Stadium just a few days after departing the Owls on Christmas Eve, with Swansea beating both Liverpool and Arsenal to drag themselves out of the Premier League relegation zone.
Despite fielding a much-changed side, partly because of injury to the likes of Leroy Fer and Wilfried Bony, Swansea settled any potential nerves with two goals in the space of as many first-half minutes.
England striker Tammy Abraham put the home side in front from close range on 18 minutes, with Nathan Dyer swiftly adding another on the break.
Dyer made it 3-0 on the half-hour with a low shot from just the edge of the penalty area, before Noor Husin got County on the scoresheet after 36 minutes.
There was, though, to be no fairytale comeback for the Sky Bet League Two promotion chasers, as Abraham scored again in first-half stoppage time.
Kyle Naughton made it 5-1 with a deflected volley following a corner on 53 minutes and Wayne Routledge curled in a sixth before midfielder Tom Carroll crashed home a well-taken seventh on 65 minutes.
Winger Daniel James marked his debut with a goal on 84 minutes to complete the route.
League One relegation battlers Rochdale beat Championship side Millwall 1-0 at Spotland to book a fifth-round home date against either Tottenham or Newport.
Captain Ian Henderson got the crucial strike, slotting in from close range after 53 minutes on what was a testing surface for both teams.
Huddersfield came from behind to win 4-1 at Birmingham after extra-time, which set up a home fifth-round tie against Manchester United.
Birmingham had taken the lead on 52 minutes when Che Adams turned on the edge of the penalty area to drill in a low shot.
Huddersfield, though, were soon level when Blues defender Marc Roberts inadvertently knocked the ball past keeper David Stockdale for an own goal following a scramble in the six-yard box.
With neither side able to conjure a winner, the tie went to extra-time, and Steve Mounie soon headed the Premier League side in front on 94 minutes.
Dutchman Rajiv van la Parra scored from a rebound to swiftly make it 3-1, before Tom Ince drove in a fourth on 106 minutes to secure the Terriers a clash against United.
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