Top seed Cilic from Croatia outplayed eighth seed Frenchman Pierre-Hugues Herbert 6-3 6-2 in 64 minutes but Anderson survived a mighty scare from a dogged Mikhail Kukushkin before edging past 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 he found a way to win.
Kukushkin was a picture of determination and perseverance as he returned almost everything, refusing to be intimidated by 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.
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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 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 break 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 blasting an easy winner on return in open court.
Kukushkin kept pushing himself but was broken in 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 the match in his favour.
Cilic will now take on a very impressive French man Gilles Simon, who ended the run of Spanish qualifier Ricardo Ojeda Lara with a 6-2 6-3 win.