Middlesbrough had knocked out Manchester United and Tottenham Hotspur on their path to the quarter-finals, but their dreams of claiming another Premier League victim were shattered early when Lukaku tapped in from Mason Mount’s cross after a swift Chelsea attack.
With a brilliant dipping effort from the edge of the box, Hakim Ziyech made it 2-0 midway through the first half, before Anfernee Dijksteel cleared off the line to deny Lukaku a second.
Middlesbrough fought hard in the second half as they attempted an unexpected comeback, with Arsenal loanee Folarin Balogun shooting wide following a promising attack and Duncan Watmore also blasting over the bar from a decent position.
Late on, Chelsea should have added a third goal when substitute Timo Werner met a free-kick, but Middlesbrough goalkeeper Joe Lumley made a great save.
Chelsea will learn who they will face in the semi-finals, which will be held at Wembley on April 16 and 17, when the draw is announced before Sunday’s quarter-final between Nottingham Forest and Liverpool. (18:30 UTC).