Lukaku restored Everton's lead in a thrilling second half at Goodison Park, two minutes after substitute Jesus Navas equalized for City on a counterattack led by Sergio Aguero.Defender Ramiro Funes Mori put Everton ahead in first-half injury time when he steered home the rebound after Ross Barkley's shot was parried out.
Lukaku, who hobbled off with an apparent injury soon after scoring, has 19 goals this season. He's failed to score in only two of his last 12 games.
The second leg is on Jan. 27.
Liverpool leads Stoke 1-0 after the first leg of the other semifinal.
Everton's home form has been patchy, with just three wins from its last 11 games at Goodison Park, but Roberto Martinez's side was impressive after weathering a good early spell by City.
And in Barkley, Everton had arguably the game's best player, regularly finding space behind City's midfield to make probing runs toward its shaky defense.
It was Barkley who instigated the opener, firing in a fierce effort that Willy Caballero — starting for City ahead of first-choice goalkeeper Joe Hart — couldn't hold. Funes Mori was on hand to turn the ball home for the Argentine's second goal for Everton.
By then, Everton had had two goals disallowed for offside — through John Stones and Lukaku — as City faded after a bright start.
Aguero squandered two chances early in the first half although it was Everton which looked more dangerous, with Barkley shrugging off Nicolas Otamendi and going one-on-one with Caballero, only to be denied by the goalie.
Everton conceded after overcommitting in search of a second goal. Aguero controlled a long clearance forward and had only two defenders in front of him. He laid in a perfectly weighted pass for Navas, who had sprinted upfield and met it with a first-time diagonal shot past Joel Robles.
Lukaku had the final say, planting home a firm header from Gareth Barry's left-wing cross — potentially setting up a Merseyside derby in the final at Wembley Stadium next month.
Everton has never won the League Cup, losing the final in 1977 and '84.
Related story
Everton crashes Man City; heaviest league defeat of Guardiola
