The hosts went into the game looking to bounce back from their Champions League exit at the hands of Sevilla in midweek, whilst Brighton were hoping a Cup run would continue to distract them from their Premier League relegation battle.
The game got off to a slow start as both sides set up to get a feel for their opponent, with a trip to Wembley at stake.
United were the first to threaten as Antonio Valencia was awarded acres of space down the right but Romelu Lukaku couldn’t quite direct his effort goalwards after getting across his marker at the near post.
That was the only real chance in the opening exchanges, before the game heated up around the 30-minute mark.
First, defender Chris Smalling saw his effort hit the outside of the post on its way behind after Juan Mata’s corner dropped to his at the near post, before Lewis Dunk could only plant a header within the reach of Sergio Romero at the other end after escaping the attentions of Scott McTominay.
With 37 minutes on the clock Brighton were made to regret that miss as the hosts found the opener.
Luke Shaw rolled the ball into the path of Nemanja Matic out on the left and he crossed first time to the far post where Lukaku powered home a header to put United ahead.
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That goal put the hosts in the lead at the break, though they were second best for large spells during the second half.
The visitors had the bulk of the possession and the chances in the second period, with Jurgen Locadia forcing two decent stops from Romero, before wasting a promising opening by heading over Solly March's inviting ball at the far post.
Despite being desperate to get back into the game Brighton boss Chris Hughton refused to go gung-ho, with all his substitutions like-for-like.
In the end, it didn't matter for the visitors as Matic made the game safe for United with just 7 minutes to play. Ashley Young - introduced at half-time in place of Luke Shaw - sent in a perfect free-kick from the left, with Matic rising to nod home at the far post and help United over the line.
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Pascal Gross twice went close in added-on time but it was too little too late for Brighton, whose focus will now return to their league form.
United, on the other hand, join Tottenham Hotspur in the hat for the semi-finals.