The young Reds took the lead twice before being pegged back by the visitors, then Ben Brereton scored twice in the opening period of extra-time to seal the win.
Toby Edser’s superb free-kick gave The Reds the lead and Virgil Gomis thought he had found the winner with a curling effort, but Ben Morris and Andre Dozzell pegged them back before Brereton struck the decisive efforts.
Forest started well, looking composed on the ball and knocking the ball around in the first few minutes, looking to use the pace of Brereton and Gomis in behind the Ipswich back-line.
But the visitors looked threatening themselves as a couple of dangerous crosses caused some nervy moments in The Reds’ defence.
The chances in the opening half-hour were limited and it took a moment of brilliance from Edser to break the deadlock on 28.
Gomis did well to get in front of his man and win a free-kick around 25-yards from goal, and Edser stepped up superbly to curl the ball over the wall and into the far corner of the net.
Flynn Downes had a long-range effort for the visitors which was straight at Jordan Wright, but otherwise the Forest keeper was untested as The Reds went into the break 1-0 up.
Forest made a quick start to the second-half, with Danny Preston’s surging run down the left flank creating an opportunity for Jake Taylor, but his shot from eight yards went over and Brereton skewed an effort wide after a quick free-kick by Taylor set him free.
But it was Ipswich who found they next goal as Ben Morris levelled on 53. A neat pass by Andre Dozzell found Morris in space inside the area, and the Ipswich number nine fired a left-footed effort high into the net past Wright.
Forest tried to push forward to regain the lead, with several through ball attempts for Brereton cut out by last-ditch blocks and challenges. Kyle McClean tested Harry Wright in the Ipswich goal on 67, as his low shot from the edge of the box was tipped around the post.
The Tractors Boys also posed a threat going forward as Conor McKendry stole the ball off Preston by the touchline on 70, and his shot could only be parried by Wright.
Gomis had a great chance to put Forest ahead moments later as he latched onto a threaded ball by McClean. His first effort bounced off keeper Wright and back into his path, but he couldn’t find the target from a tight angle as he hit his shot into the side-netting.
Gomis nearly made amends for the miss as he stung the palms of Wright on 75, hitting a half-volley from just inside the area which the keeper beat away but he was not to be denied as he restored the lead with six minutes left.
The ball broke to him on the edge of the Ipswich box and he curled an effort over Wright and into the top corner to send the young Reds wild.
But Ipswich responded again and Dozzell produced almost a carbon-copy of Edser’s first-half free-kick to make it 2-2.
Town almost found the winner moments later. Firstly McKendry was denied by a superb stop by Wright who flung to his left to tip his looping shot over, and then Corrie Ndaba fired agonizingly wide from the corner.
That was to be the final action of normal time as the tie headed to extra-time.
And it didn’t take long for The Reds to restore their lead as they were back in front on 96 minutes. Brereton picked up the ball in midfield and played it out wide to Gomis. Brereton ran through the middle and Gomis played a perfectly weighted through-ball into his path and he calmly slotted home for 3-2.
And the pair combined again to create daylight on 100 minutes. Gomis was alert to steal the ball on halfway and he played it off for Brereton to run in on goal and toe-poke past Wright.
The fourth goal seemed to knock the stuffing out of Ipswich, but Dozzell flashed an effort wide on 107 as their tired legs tried to find a way back into the match, and McKendry hit an effort over the bar on 113.
But they could not find a way through as Forest controlled the closing stages of the game well and secured their place in the fourth round.
Forest: J. Wright, Charlesworth, Harbottle (Jemson 91’), Re (Appiah 82’), Preston, Taylor (En-Neyah 67’), Ahmedhodzic, Edser, McClean, Gomis, Brereton
Subs: Shelvey, C. Smith
Goals: Edser 28’, Gomis 84’, Brereton 96’, 100’
Yellow cards: Re 62’, Taylor 63’, En-Neyah 87’, Ahmedhodzic 90’, McClean 90’
Ipswich: H. Wright, Woolfenden, Ndaba, Downes, Cathline (Cahill 88’), Webber, McKendry, Dozzell, Morris (Daniels 87'), Folami, Meldrum (McGavin 60’)
Subs: Clements, Hayes
Goals: Morris 53’, Dozzell 88’
Yellow cards: Webber 27’, Woolfenden 57’