Middlesbrough defeated Manchester United on penalties in the fourth round and scored an extra-time winner to beat Tottenham Hotspur in the fifth round, but Chris Wilder's side came unstuck against Chelsea on Saturday.
Middlesbrough were given raucous support inside the Riverside Stadium but never really looked capable of troubling the visitors who were always in control.
Chelsea, who are the subject of several bids to buy the club, are operating under a special licence from the British government after Russian owner Roman Abramovich was sanctioned following Russia's invasion of Ukraine.
Only a few hundred Chelsea fans were inside the stadium -- a consequence of restrictions on them selling tickets -- but they could at least celebrate their team reaching the semis for the fifth time in six seasons.
Since Abramovich announced that he was putting the European champions up for sale on March 2 after 19 trophy-laden years, it has been very much business as usual on the pitch.
They have won all six fixtures since, the latest win coming the day after a deadline for bids to buy the club.
Despite the distractions, manager Thomas Tuchel has remained focused on winning games, and the performance of his players on Saturday was clinical.
"We were very focused and very serious. We showed the quality at decisive moments," the German said.
"The effort was huge and we never let them get a foot into the door after the two early goals. It was a deserved win."
"Nothing we can do will change the situation and we accept that it's not in our hands, but what is in our hands is setting an example and staying focused."
Chelsea had attracted criticism in midweek for asking for the match to be played without fans to ensure "sporting integrity" in light of them not being able to sell tickets.
They later retracted that request and the Riverside was rocking before kickoff.
But when Mount was allowed to advance down the left and then delivered a slide rule cross to allow Lukaku to tap home, the odds always looked stacked against Middlesbrough.
When Ziyech's shot then flew past Middlesbrough goalkeeper Joe Lumley the hosts had a mountain to climb.
Chelsea could have added to their lead with Lukaku denied by a goalline clearance and Lumley doing well to keep out Timo Werner's late effort, while the hosts rarely had a sniff of a chance.
It was Middlesbrough's first home defeat in 10 games in all competitions, and their focus will now return to trying to make the promotion play-offs.
For Chelsea, they are still chasing silverware on two fronts after reaching the Champions League quarterfinals in midweek.
"We really have that winning mentality within the group. Any competition, any game, we want to win, and we want to keep that going," Mount said. "It's been a tough couple of weeks but for us, as a team, we're just focused and ready to play."
Information from Reuters contributed to this report.