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Goals for Marcus Rashford, Wout Weghorst – his first for the club – and Bruno Fernandes secured victory for the visitor at the City Ground.
After dropping points in back-to-back Premier League games against Crystal Palace and Arsenal, it marked a return to winning ways for Erik ten Hag's men.
For Steve Cooper's hosts though, it marks a bitterly disappointing result, with any chance of a comeback at Old Trafford now likely scotched.
United needed just six minutes to carve Forest open, with Rashford afforded space to scamper down the left wing and squeeze between two defenders before beating Wayne Hennessey at the near post.
Forest thought they had equalised with a superb breakaway finished by Sam Surridge in the 23rd minute, only for the VAR to deem the striker offside.
The host looked to be growing into the game after that but received a sucker punch on the stroke of half-time when Weghorst turned in after Antony's effort was parried.
Christian Eriksen nearly added a third nine minutes after the interval, his looping attempt hitting the crossbar.
With time running out, Forest called upon former United midfielder Jesse Lingard from the bench in the hope of salvaging something for the second leg.
But the result was put beyond doubt in the 89th minute, Fernandes drilling into the bottom-left corner to surely send United to Wembley.