The Croat was confirmed as joining the likes of Roger Federer and Novak Djokovic in next month's finals in London on Saturday, completing a productive few weeks for the 26-year-old after he clinched the US Open title with a win over Kei Nishikori in September.
"This was one of my childhood dreams," said Cilic. "It happened this year and I'm extremely happy about this success. It's another achievement in my career.
"The ATP World Tour Finals is for sure one of the most interesting tournaments of the year. It is watched all over the world and brings a big attendance to all the matches. It will be a big honour for me to participate there."
Seeded second for the ATP 250 hard-court event in Moscow, Cilic had to come from a set down to see off 2013 runner-up Kukushkin 4-6 6-3 6-2.
Cilic lost the first set after Kukushkin sealed a break in game nine, and the Kazakh broke the world number eight in the opening game of the second to raise hopes of victory.
Cilic hit back immediately, however, before forcing a decider with a second break in game eight.
From that point, the Croat was dominant in the final set, breaking his unseeded opponent twice more to move into his 21st ATP Tour final.
He will face Roberto Bautista-Agut on Sunday, after the Spaniard overcame Ernests Gulbis 6-4 6-4.
Fifth-seed Bautista-Agut made the ideal start when he raced into a 40-0 lead against the Gulbis serve in the very first game of the match.
Although third seed Gulbis saved two of those break points, he was unable to rescue the third as his opponent seized the early initiative.
Bautista-Agut then held his serve throughout the remainder of the set to move one step closer to the final.
Gulbis was forced to save two more break points in game five of the second set, but he was unable to prevent Bautista-Agut from forging ahead in the seventh as the Spaniard moved 4-3 to the good with a break.
The Latvian rescued a match-point at 5-4 40-30 down in the ninth, but Bautista-Agut was not to be denied as he then served out to reach his fourth ATP final and third in 2014.