Nagoya Grampus, however, bowed out of the season-closing tournament with a 4-3 loss on penalties to Yokohama F. Marinos.
Cerezo made it to the last four for the first time since the 2005 edition after Shimizu goalkeeper Kaito Yamamoto missed his side's sixth penalty.
Midfielder Kim Bo Kyung put the home team ahead in the 11th minute, but the visitors levelled in the 23rd minute thanks to Shinji Ono's penalty after Taikai Uemoto handled Naohiro Takahara's shot.
Three minutes into extra time Cerezo regained the lead when Kiyotake cut into the box to slot home, and the Japan midfielder was twice denied by Yamamoto before Shimizu substitute Toshiyuki Takagi equalised.
With the scores tied at 5-5 in the shootout, Rui Komatsu scored for Cerezo before Yamamoto's attempt hit the woodwork.
Cerezo boss Levir Culpi said: "I was convinced we could win the game as all our players looked so confident before going into the shootout.
"We actually created more big chances than Shimizu did. We deserved the win today."
Cerezo will meet Division Two winners FC Tokyo at home in the semi-finals on December 29.
The west Tokyo team beat Urawa Reds 1-0 thanks to Naohiro Ishikawa's 20th-minute winner.
Grampus, who played without Joshua Kennedy, Marcus Tulio Tanaka and Shohei Abe due to injury and the suspended Hayuma Tanaka, were held to a scoreless draw over 120 minutes by Marinos.
In the shootout, Nagoya's final two takers Keiji Yoshimura and Danilson failed with their attempts, while Marinos were successful with all four kicks.
Marinos will next play Kyoto Sanga on Thursday at the National Stadium in Tokyo. Kyoto beat Division Two counterparts Shonan Bellmare 1-0, Dutra scoring the game's only goal.