The Swans opened the scoring on 18 minutes when Tammy Abraham tapped home Luciano Narsingh's cross at the near post and added a second a minute later when Nathan Dyer slotted past Adam Collin after latching on to Abraham's defence-splitting ball.
Carlos Carvalhal's charges added a third on the half-hour mark when Narsingh rolled the ball into the path of Dyer, who swept home for his second of the game, although County pulled a goal back on 36 minutes through a long-range Noor Husin effort.
Any hope that the visitors had of mounting a comeback was snuffed out on the stroke of half time when Kyle Bartley found Narsingh out wide, who in turn crossed for Abraham to sweep home first time.
Goal number seven came on 65 minutes when Tom Carroll fired a long-range effort into the top corner of Collin's net before the icing on the cake came six minutes from time as Daniel James volleyed home from Abraham's knockdown.
The result means that Carvalhal will return with Swansea to former club Sheffield Wednesday at Hillsborough in the fifth round.
SWANSEA CITY (4-4-1-1): Nordfeldt; Naughton, van der Hoorn (Maric 78'), Bartley, Roberts; Narsingh, Carroll, Clucas (Ki 45'), Routledge; Dyer (James 62'); Abraham
NOTTS COUNTY (4-4-1-1): Collin; Hunt, Duffy, Brisley, Dickinson; Grant, Virtue-Thick, Husin (Milsom 62'), Hawkridge; Alessandra (Forte 62'); Stead (Smith 65')