Ole Gunnar Solskjaer's side dominated for long periods of their quarter-final at Goodison Park, but it took until the 88th minute before Edinson Cavani broke the deadlock.
The former Paris Saint-Germain striker's fourth goal for United since joining on a free transfer in October came at the same venue where he opened his account for the club last month. Anthony Martial bagged United's second goal in stoppage time.
After a rocky period that raised questions about Solskjaer's future following their exit from the Champions League, United have recovered their equilibrium. They are third in the Premier League and have won their last three matches in all competitions. United last lifted silverware when they won the Europa League in 2017 and Solskjaer says they are "desperate" to end that barren run. The Norwegian, who won nine major trophies as a player with United, took charge at Old Trafford in 2018 and saw his side lose in three semi-finals last term.
He now has a chance to go one better this season after United joined holders Manchester City, Tottenham and second-tier Brentford in the League Cup semi-finals.
December 23, 2020
Everton made only three changes from the 2-1 win over Arsenal, with Dominic Calvert-Lewin and Richarlison both starting in attack. In contrast, Harry Maguire and Bruno Fernandes were the only United players to retain their places after the 6-2 rout of Leeds at the weekend, with Paul Pogba returning to the starting line-up. But it was United who dominated and Robin Olsen kept out Cavani's close-range effort before Mason Greenwood's header clipped the far post.
Solskjaer sent on Marcus Rashford and Martial in the second half and it proved an inspired move. However, the game could have been very different had Edinson Cavani been sent off in the 52nd minute. Replays showed he grabbed Yerry Mina around the throat, but with no VAR, the Uruguayan escaped punishment.
With just two minutes left before a penalty shoot-out, Cavani put United ahead. Martial slipped a pass to Cavani, who drove a superb strike past Olsen from the edge of the area. Martial then slotted home with virtually the last kick after Everton were caught throwing too many players forward.
Man Utd will host Man City in the semi-finals while giantkillers Brentford go to Tottenham. The ties will take place on the week commencing 4th January.