Top seed Cilic from Croatia outplayed eighth seed Frenchman Pierre-Hugues Herbert 6-3 6-2 in 64 minutes while Anderson survived a mighty scare from a dogged Mikhail Kukushkin before edging him 6-7(3) 6-4 6-2.
The US Open finalist from South Africa had to strategise and re-strategise against the Kazak in an intense quarterfinal, which lasted two hours and 30 minutes.
His big serve, with which he fired 24 aces, was put to test by the Kazakh, but eventually, the tall South African found a way to win.
"He is one of the best returners in the world. You hope to hit aces and get some free points but you have to be ready for balls to come back against him. He was staying back at the baseline, so I served and volleyed and executed nicely. He has great defence so used a few drop shots as well," Anderson said after the match.
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Anderson recently hired a new coach in Brad Stine, who also helped former world number one Jim Courier in the mid- 90s.
It was the third meeting between the two players and after the latest win, Anderson led 2-1.
Kukushkin was a picture of determination and perseverance as he returned almost everything, refusing to be intimidated by the six feet eight inch Anderson.
The South African came out blasting big serves in the range of 200 but Kukushkin's anticipation was so spot on that he got himself positioned nicely to return everything.
Initially, the South African hardly lost a point on his serve but as the match wore on, it was Kukushkin who started to dominate. His consistency with returns had Anderson gasping. His was not a big serve like Anderson's but his returns made a huge difference.
Kukushkin was still not dominating but trying very hard to stay in the game as his execution was not proper. Both of them continued to test each other and soon the South African got the break to take a 5-3 lead.
However, Kukushkin with his excellent court coverage found a way to get the break back, converting his fourth chance in the ninth game and saved a set point in the next to make it 5-5.
Anderson saved two set points but netted a backhand on the third as Kukushkin pumped his fist in the air to celebrate.
His confidence level going up, Kukushkin's serve also got better as the match progressed. Neither of the two players faced a beak point in the first six games.
Anderson found a service return winner in the seventh to earn his first break chance but Kukushkin saved that with a deep serve and blasted an easy winner on return in open court.
Kukushkin kept pushing himself but was broken in the third game and was unable to convert two chances in the fourth to fall behind 1-4 in the decider.
From there, it was easy for the world number 14 to close out the match in his favour.
Cilic will now take on the very impressive Frenchman Gilles Simon, who ended the run of Spanish qualifier Ricardo Ojeda Lara with a 6-2 6-3 win.