The fifth-round contest looked to be heading for extra-time, but Casemiro's 89th-minute header sent the 20-time English champions into the final eight, where they will take on Liverpool at Old Trafford.
Sofyan Amrabat, Scott McTominay and Antony came into the Man United side following the home loss to Fulham in the Premier League on Saturday, with Harry Maguire, Kobbie Mainoo and Omari Forson dropping out.
As for Forest, Matt Turner, Harry Toffolo, Ryan Yates and Divock Origi came into the XI following the 4-2 defeat at Aston Villa in the league last time out.
Man United came close to making the breakthrough in the fourth minute when a well-worked corner saw Bruno Fernandes deliver into Antony after combining with Alejandro Garnacho, but the Brazil international's powerful strike came off the crossbar in a let-off for the home side.
The visitors had another big chance in the seventh minute when Diogo Dalot delivered onto the head of McTominay, but the Scotland international headed straight at Turner from inside the box.
Down the other end, Man United goalkeeper Andre Onana was forced into action to keep out a strike from Origi after the ex-Liverpool attacker's pace had taken him away from Amrabat, who was playing at left-back.
Taiwo Awoniyi then had two chances in quick succession, but both of his efforts were blocked by Victor Lindelof, with Forest starting to find space in the final third of the field.
Marcus Rashford then fired over the Forest crossbar in the 17th minute, before Onana made a smart save down the other end to keep out a strike from Awoniyi.
Origi tried his luck again in the 29th minute, lifting the ball over the crossbar, with both sides having their moments in the final third of the field.
McTominay had another headed chance in the 33rd minute after an excellent cross from Dalot, but the Scotland international was again denied by Turner, before Neco Williams struck over the crossbar with a rasping effort down the other end, with the two teams level heading into the final stages of the first period.
Awoniyi fired wide of the Man United goal in the closing minutes of the opening 45 minutes, before Casemiro missed the target at the conclusion of a smart passing move from the visitors.
Onana made a smart save to keep out a strike from Awoniyi early in the second period, with Forest starting on the front foot; Man United's goalkeeper was again called into action in the 51st minute to deny Origi, who smashed one towards the back of the net.
Turner made a strong stop to keep out a strike from Antony in the 58th minute, before Man United had a penalty shout turned down, with Rashford going down inside the box under a challenge from Murillo.
Fernandes sent one just over the Forest crossbar in the 63rd minute with an inventive effort, before the Portugal international had a strike blocked wide of the home side's post by his own teammate McTominay three minutes later.
Turner denied Fernandes in the 72nd minute during a period when Man United were on top, with the Red Devils starting to find their passing rhythm in the final third of the field.
Amad Diallo had a half-chance at the far post in the 76th minute after a Garnacho effort deflected kindly for him, but the Ivorian was unable to make the most of the opportunity, before Onana did well to keep out a long-range effort from Murillo down the other end.
Rashford sent a wild effort wide of the Forest goal in the 82nd minute, before Casemiro just missed the target two minutes later, with extra-time looming large at the City Ground.
Man United made the breakthrough in the 89th minute of the contest, though, with Casemiro heading a free-kick delivery from Fernandes into the back of the net from close range.
There was a lengthy VAR check for offsides on both Raphael Varane and Casemiro, but eventually it was cleared, and the goal proved to be enough for Erik ten Hag's side to book their spot in the quarter-finals of the FA Cup, with Liverpool set to be their opponents in a glamour affair.