Middlesbrough had eliminated Manchester United and Tottenham on their way to the quarter-finals, but their hopes of claiming another Premier League victim were dashed early when Lukaku tapped in from Mason Mount’s cross following a quick Chelsea counter.
Hakim Ziyech made it 2-0 midway through the first half with a superb dipping effort from the edge of the box, before Anfernee Dijksteel cleared off the line to deny Lukaku a second.
Middlesbrough showed plenty of fight in the second half as they attempted an unlikely comeback, with Arsenal loanee Folarin Balogun shooting wide after a promising counter, and Duncan Watmore also firing over the bar from a good position.
Chelsea could have added a third late on when substitute Timo Werner met a free-kick, but Middlesbrough goalkeeper Joe Lumley saved well.
When the draw is made before Sunday’s quarter-final between Nottingham Forest and Liverpool, Chelsea will find out who they will play in the semi-finals, which will be played at Wembley on April 16 and 17. (18:00 GMT).