A hat-trick from Tyrese Campbell inspired The Potters to victory against 10-man Forest who had Choz Charlesworth sent off for two yellow cards.
Luke McCormick struck in the first-half for The Reds after Campbell had put Stoke 2-0 up, but second-half goals by Campbell and Harvey Read completed the victory and sent Stoke through to the fifth round.
The home side exerted some early pressure on the Forest goal with Campbell going close having worked his way to the byline and a superb block by Charlesworth denying Jordan Greenidge inside the opening four minutes.
Ben Brereton had a superb chance moments later, cutting inside two defenders from the left-hand side but dragging his shot just wide from 10 yards but it was Campbell who broke the deadlock on seven minutes.
The forward drifted inside one challenge before firing low past Jordan Wright in the Forest goal to give the Potters a 1-0 lead and Campbell doubled the advantage on 18.
Campbell capitalised on a mistake by Forest defender Riley Harbottle who was caught on the ball to shoot low past Wright and give The Reds a mountain to climb.
Forest started that ascent as a fantastic McCormick goal pulled one back on 36 minutes. Jake Taylor picked the ball up in the centre circle and McCormick made a great run down the right-hand side, bearing down on goal and firing a great effort past Mate Deczki into the far corner.
The Reds had grown into the match after a nervy opening, but it was Stoke who almost found the next goal as Greenidge shot just wide from the right of the penalty box, and Forest had a good chance of their own just before half-time with a superb stop by Deczki denying Brereton from inside the six-yard box.
Forest started the second half well and had a lot of the ball but the first clear-cut chance of the half fell to Greenidge on 58 minutes. The Stoke striker skipped past the challenge of Ahmedhodzic inside the area but was thwarted by a smart stop by Wright.
The Reds made their task harder just before the hour mark as Charlesworth was sent off for a second yellow card. The full-back was booked for two late challenges inside the first 15 minutes of the second half and shown his marching orders by referee Paul Graham.
The sending off took the wind out of Forest's sails and Stoke found the third goal on 70 minutes as Campbell completed his hat-trick. The forward latched onto a through-ball and slotted an effort past Wright from 12 yards to create daylight between the two teams.
Campbell nearly added his and Stoke's fourth on 76 minutes as Wright denied him from close range, but from the resulting corner the home side did find the net as substitute Read prodded home after Wright parried an initial effort.
Stoke were able to see out the match and ensure that it would be they who progressed in this season's FA Youth Cup, while Forest showed plenty of endeavour against a strong Premier League outfit.
Forest: J. Wright, Charlesworth, Harbottle, Ahmedhodzic, Preston, McClean, Edser, McCormick (Re 62'), Taylor (Smith 62'), Gomis (En-Neyah 46'), Brereton
Unused subs: Shelvey, Jemson
Goal: McCormick 36'
Bookings: Brereton 25', Charlesworth 48', 59'
Red card: Charlesworth 59'
Stoke: Deczki, Edwards, McJannet, Cherdieu (Read 46'), Souttar, Silva, Dunwoody (C), Sorensen, Greenidge, Campbell (Szereto 78'), Twyford
Unused subs: Allen, Balde, Da Silva
Goals: Campbell 7', 18', 70', Read 76'
Bookings: Cherdieu 36', Dunwoody 45', Sorensen 52'