Weghorst added to an opener from Marcus Rashford before Bruno Fernandes struck in the final exchanges, as the 20-time English champions secured a three-goal success over Steve Cooper's side at the City Ground.
Man United, who are chasing their first trophy since the Europa League in 2017, will welcome Forest to Old Trafford for the second leg next Wednesday, with the winner advancing to the final to take on either Newcastle United or Southampton.
The visitors had a half-chance inside the opening two minutes when a mistake from Joe Worrall presented Fernandes with the ball inside the Forest box, but the attacker's low cross, which was towards Weghorst, was hooked clear by Scott McKenna.
The Red Devils did manage to take the lead in the sixth minute, though, and it was the in-form Rashford, who powered his way into the Forest box before smashing the ball past Wayne Hennessey at his near post.
Antony then bent one just wide of the Forest post in the eighth minute before just placing one over the crossbar four minutes later, with Erik ten Hag's side starting the match firmly on the front foot.
Forest's threat was mainly coming on the break through Morgan Gibbs-White and Brennan Johnson, and it was through that avenue that their leveller almost arrived, with the former finding Sam Surridge, who picked out the bottom corner with a brilliant finish.
However, VAR came to Man United's rescue, with Surridge just in an offside position when the pass was made.
A mistake from Victor Lindelof then caused panic inside the away team's penalty box, but Lisandro Martinez came to his teammate's rescue, turning the ball behind for a corner, which led to David de Gea making a fine save to keep out Gustavo Scarpa's effort.
De Gea then had to keep out a strike from Johnson in the 28th minute, before the Forest forward fired over the crossbar two minutes later after working his way to the edge of the Man United box.
Hennessey did well to deny Antony in the 32nd minute after the Brazilian had been released into the Forest box by a clever flick from Fernandes.
Scarpa missed the crossbar with a long-range attempt late in the first period, but it was Man United that scored the game's second goal in the 45th minute, with Weghorst converting from inside the box after Hennessey had saved a powerful effort from Antony.
Weghorst had a half-chance early in the second half when Antony fed his teammate inside the Forest box, but McKenna was across to challenge at the vital moment.
Fernandes then fired well wide of the target after meeting a cross from Tyrell Malacia, with the away side threatening to take the tie away from their opponents, but Forest were having their moments, and De Gea had to get down to keep out an effort from Danilo in the 53rd minute.
Christian Eriksen hit the crossbar with a brilliant strike in the period that followed, before Hennessey dived across his goal to keep out a powerful drive from Rashford.
Gibbs-White then just missed the Man United goal in the 59th minute after breaking into the penalty box before attempting to pick out the top corner, before Scarpa had a goal-bound strike blocked by Aaron Wan-Bissaka.
Jesse Lingard nodded wide of the target against his former club in the 74th minute, before Fred lifted one over the crossbar from distance down the other end; Weghorst then headed wide of the Forest post after being found by Facundo Pellistri, with the latter making his second appearance for the club.
Pellistri had another half-chance late on after being fed by Fred, but Hennessey made the save; Man United were not finished, though, and they scored a third in the final moments when Fernandes smashed the ball into the bottom corner after being found by Anthony Elanga.
Next up for Man United is an FA Cup fourth-round contest with Reading on Friday night, but Forest will not be in action until the second leg of this semi-final, as they were knocked out of the FA Cup by Blackpool.