The 26-year-old Croatian proved too strong for the experienced Spaniard, ranked 21st in the world, who was making his second appearance at the Kremlin Cup after a 12-year hiatus.
World number eight Cilic hit 10 aces and converted four break points on his way to a victory that kept him on course for his fifth final of the year.
"Today's performance was at a higher level than yesterday - my movement was better, as were my ground shots and especially my serve," Cilic told reporters.
"I was hoping to play better because Robredo is very difficult opponent, dangerous from the back of the court. We had to play games much longer than I expected. I'm really happy I could win in two sets."
The tournament favourite will now face 26-year-old Kazakh Mikhail Kukushkin in the semi-finals.
"I saw his match with Mikhail Youzhny and Kukushkin showed high quality," Cilic said.
"He is the kind of the player that likes to be aggressive. It's going to be different match than today's, but I have to keep my level and control the game better than him."
Over a decade ago, Robredo did not enjoy the happiest of times in Moscow, crashing out to Paul Henri Mathieu. He would endure similar misfortune this time around.
The US Open winner hit ten aces and converted five break points on his way to victory and now is looking to reach his fifth final of the year.
"I think today's performance was at a higher level than yesterday - my movement was better, as were my ground shots and especially my serve", said Cilic.
"I was hoping to play better, because Robredo is very difficult opponent, dangerous from the back of the court. We had to play games much longer than I expected. I'm really happy I could win in two sets."
The main favourite to win the Kremlin Cup will now face 26 year-old Kazakh Mikhail Kukushkin in the semi-finals.
"I saw his today's match with Mikhail Youzhny, and Kukushkin showed high quality. He is the kind of the player that like to be aggressive. It's going to be different match than today's, but I have to keep my level and control the game better than him," Cilic concluded.
(Reporting by Dmitriy Rogovitskiy, editing by Ed Osmond)
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