First-half goals from Liam Cullen and Matt Grimes put Swansea in charge of the tie but Forest began the second half in improved fashion and deservedly reduced the arrears when Anthony Knockaert lashed home his second goal of the season.
A controversial penalty decision allowed Grimes to score his second of the afternoon before further goals from Cullen and substitute Oli Cooper confirmed Swansea's place in the fifth round of this season's competition.
Chris Hughton made ten changes from the side beaten by Middlesbrough on Wednesday night with only Joe Worrall keeping his place in the starting line-up while there were places in the side for Fouad Bachirou and Luke Freeman on their return from their respective injuries.
The Reds made a bright start at the Liberty Stadium as Knockaert curled an effort over Freddie Woodman's crossbar in the seventh minute of the contest but they were behind soon after that. Swansea took the goal-kick quickly and broke down the other end of the field and Wayne Routledge's cross was perfect for Cullen to volley smartly beyond Jordan Smith.
Steve Cooper was forced into an early change as Routledge limped off to be replaced by Jamal Lowe and the former Wigan striker almost had an instant impact as he was set through on goal but Smith came off his line well to smother the ball and clear the danger.
The home side doubled their lead just before the half-hour mark as a corner was cleared out only as far as Grimes and the Swansea skipper hammered a terrific 25-yard strike which beat Smith all ends up and thundered into the back of his net to put The Swans firmly in charge of the tie.
Forest needed a much-improved start to the second half to get back into the tie and they came storming out of the blocks and halved the deficit. Joe Lolley was brought down inside the Swansea penalty area but referee Kevin Friend played the advantage and allowed Knockaert to lash the ball into the back of the net.
Just as The Reds thought they had grabbed a route back into the contest, it was snatched away from them five minutes later. Swansea full-back Ryan Manning went down on the very edge of the penalty area under Worrall's challenge, Friend pointed to the spot and Grimes sent Smith the wrong way for his second and Swansea's third goal of the game.
It seemed like Forest were still reeling from that decision when they conceded a fourth goal on 67 minutes. Swansea worked the ball well into a crossing position on the left-hand side and as the ball was played in, Cullen was there to sweep the ball beyond Smith and seal the home side's place in the fifth round.
Worse was to come for The Reds with six minutes of normal time to play when Cooper, who had come on for recent new signing Conor Hourihane, drilled the ball underneath Smith's body at the near post to ensure Forest would leave South Wales on the end of a disappointing defeat.
Forest: Smith; Jenkinson, Worrall (c), Figueiredo, Bong (Blackett 76'); Bachirou (Cafu 46'), Mbe Soh; Lolley, Freeman (Guerrero 46'), Knockaert (Mighten 76'); Taylor
Subs not used: Samba, Christie, Grabban, Ameobi, Dawson
Goal: Knockaert 56'
Booking: Jenkinson 10'
Swansea: Woodman; Latibeaudiere, Bennett, Guehi (Bidwell 65'); Roberts (Garrick 76'), Hourihane (Cooper 65'), Grimes (c) (Fulton 65'), Manning; Dhanda; Routledge (Lowe 20'), Cullen
Subs not used: Webb, Ayew, Naughton, Cabango
Goals: Cullen 7', 67', Grimes 29', (pen) 61', Cooper 84'
Referee: Kevin Friend