United scored at the beginning and end of the first half to put themselves in control, with Marcus Rashford netting a solo goal to continue his red-hot form and Wout Weghorst converting a simple rebound to get his first goal for the club since his loan move from Burnley.
Forest, seeking their first League Cup final appearance since 1992, gave a strong account of themselves but were unable to make the most of several promising attacks. And an 89th minute goal from Bruno Fernandes means next week's return leg at Old Trafford is surely no more than a formality for Man United to secure a trip to Wembley.
"I'm happy with the performance, over 90 minutes we controlled the game," United coach Erik ten Hag said. "There was one moment which could have changed the game and we have to be aware of that and improve.
Ten Hag's determination to make the most of a chance to win a trophy in his first season in charge at Old Trafford was evident in another strong starting lineup that included Casemiro returning from the suspension he served in Sunday's Premier League defeat to Arsenal.
It took just six minutes for the visitors' to go in front, with Rashford once again finding the net. This time the goal was all the England forward's own work as he drove forward down the left from inside his own half before driving inside between two Forest defenders and firing in at the near post with his left foot.
The goal was Rashford's 10th in 10 games since the World Cup, the most by any player in Europe's top five leagues.
"I am happy with Marcus' performance and his development," Ten Hag said. "From the start of the season he is growing and keeps doing that.
"We have to keep his process going and if he is in that mood and spirit I think he is unstoppable. It is up to the team to get him in the rigjht positions."
But Forest were undeterred by that early concession and on several occasions came close to an equaliser. Brennan Johnson's pace was a consistent threat to the United defence, allowing him to force a low save from David de Gea and later blasting over the bar when his composure deserted him at the last.
The hosts thought they had pulled level in the 23rd minute when a lightning break culminated in Sam Surridge finding the net but a VAR check ruled he had been offside.
Then right before the half-time interval, United grabbed a potentially crucial second. Antony, who had already seen one good chance denied, forced Forest goalkeeper Wayne Hennessey to block his well-struck volley, allowing Weghorst to tap-in to open his United account.
It was a similar story in the second half. United went close to making it 3-0 when Christian Eriksen curled a first-time shot against the crossbar. At the other end, Forest were again left to rue a lack of clinical touch when the lively Morgan Gibbs-White fired wildly off target after winning the ball just outside the box.
Fernandes' late strike, driving in low from just inside the box, was perhaps cruel on Forest, but will surely see Man United head to Wembley as they seek their first major trophy in six years.
"Obviously, it was a disappointing scoreline," Forest coach Steve Cooper said. "The last thing you want to do is go to Old Trafford with this result.
"Obviously, Rashford is one of the in-form players in the country, but for us to allow him to do what he did is not good enough."
Newcastle United won 1-0 at Southampton in the first leg of the other semifinal.
Information from Reuters was used in this report.