The Red Devils entered the contest with a 3-0 advantage from the first leg, and they finished the job in front of their own supporters, with Anthony Martial and Fred on the scoresheet at Old Trafford, as the 20-time English champions ran out 5-0 winners on aggregate.
Man United will take on Newcastle United in the final of the competition on February 26, with Erik ten Hag looking to win his first trophy since his arriving as head coach.
Ten Hag took no chances with his team selection, with Raphael Varane, Lisandro Martinez, Luke Shaw, Casemiro, Bruno Fernandes and Antony among those to feature in a strong first XI.
Forest had nothing to lose and everything to gain, and they started on the front foot, with Emmanuel Dennis, who was brought into the side as a late replacement for the injured Jesse Lingard, carrying the fight.
Man United had a shout for a penalty in the 17th minute when Alejandro Garnacho went to ground inside the box under a challenge from Neco Williams, but the referee was not interested and VAR agreed.
The game's first attempt on target came in the 18th minute from Antony, but it was a comfortable save from Wayne Hennessey between the away side's sticks.
The first serious chance of the match came on the half-hour mark when Shaw delivered a dangerous free kick onto the head of Casemiro, but the Brazilian's effort was kept out by Hennessey.
Forest had a brilliant opportunity of their own in the 34th minute when Brennan Johnson found himself free inside the Man United box, but he fired straight at Tom Heaton.
Antony then struck one just over the crossbar four minutes later, before Forest had their best chance of the half; Dennis struck towards goal following a break down the right, and the ball looked to be finding the bottom corner, only for Sam Surridge to block his teammate's effort.
The final opportunity of the first half fell to Man United, with Wout Weghorst meeting a cross from Casemiro, but the striker's header hit the outside of the post.
The home side had another shout for a penalty in the 55th minute when Scott McKenna kicked Weghorst inside the Forest box, but neither the referee nor VAR were interested in what was a controversial decision.
Garnacho brought a smart save from Hennessey just before the hour, and Man United then made three changes, with Marcus Rashford and Martial entering the field alongside Jadon Sancho, as the attacker featured for the first time since the end of October.
Casemiro headed just wide of the Forest post 25 minutes from time, before Rashford had a shot blocked inside the box, with the score remaining goalless at Old Trafford.
Man United finally made the breakthrough in the 73rd minute, though, with Martial finding the bottom corner with a smart finish after initially looking to slide a pass into Rashford.
The home side then doubled their lead in the 76th minute through Fred, with the Brazil international finishing from close range after Rashford had put it on a plate for his teammate.
Forest came close to scoring in the 82nd minute when Danilo broke through, but Heaton was on hand to make a brilliant save before Surridge fired over from the rebound.
Fernandes fired one just wide of the Forest post in the 86th minute, with Man United looking to add further gloss to the scoreline at Old Trafford.
However, the Red Devils ultimately declared at two, with the home side comfortably booking their spot in the final of the competition courtesy of a five-goal aggregate success.
Sancho was bright on his return to the first team, and the England international will now be looking to play an important role in the final months of the 2022-23 campaign.
Next up for Man United is a Premier League clash with Crystal Palace on Saturday, while Forest will continue their league campaign against Leeds United on Sunday.