All of City's goals at Old Trafford came in a one-sided first half that highlighted the gulf which has opened up between the cross-town neighbours over recent years.
When the visitors went 3-0 up courtesy of an own-goal by Andreas Pereira in the 39th minute, memories were revived of the 6-1 win served up on this ground by Roberto Mancini's City in 2011 that signalled the shifting of the balance of football power in Manchester.
However, United managed to avoid conceding any more goals and a 70th-minute strike by Marcus Rashford gave Ole Gunnar Solskjaer's team a glimmer of hope heading into the second leg at City's Etihad Stadium on January 29.
City have won the League Cup the last two seasons and are now unbeaten in 16 straight matches in the competition, dating back to a loss at United in the fourth round in October 2016.
Aston Villa take on Leicester in the other semifinal on Thursday (NZT).
Mindful of how United tore apart his City team on the counterattack in a Premier League game won by Solskjaer's side at the Etihad last month, Pep Guardiola chose not to play a recognised striker - Sergio Aguero and Gabriel Jesus were on the bench - and flooded his midfield to outnumber the hosts in that department.
City's midfielders picked their moments to charge forward, and did so with devastating effect.
Bernardo Silva was unchallenged as he advanced toward the edge of United's penalty area and drove a fierce, curling shot into the top corner in the 17th minute, with goalkeeper David De Gea barely moving.
The Portugal playmaker then wriggled into space in the centre of the field and threaded a perfectly weighted through ball to Riyad Mahrez, who ran onto the pass, rounded De Gea and slotted the ball into an empty net in the 33rd.
United were reeling and it got even worse six minutes later. Again their defence opened up as Kevin De Bruyne went round Phil Jones with ease and fired in a left-footed shot that was saved by De Gea, only for the ball to ricochet off the shin of Pereira and rebound into the net.
With Raheem Sterling wasting two glorious chances that were teed up for him from crosses from the right, City should have gone into halftime with an even bigger lead.
Mahrez struck the outside of the post early in the second half and also brought out a good save from the feet of De Gea, before United finally got in behind City's defence to score a goal that could yet prove to be more than a consolation.
Mason Greenwood slipped a well-timed pass into the path of Rashford, who took a touch and finished low past City goalkeeper Claudio Bravo.
Away goals do not count double if the aggregate score is tied after two legs in the League Cup.