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Cilic's 6-1, 6-1, 6-1 victory gave Croatia, the 2005 champions, an unassailable 3-1 lead in the quarter-final.
Cilic had already beaten Dmitry Popko on Friday 6-2, 6-1, 6-2.
On Saturday, Ivan Dodig and Nikola Mektic beat the Kazakh pair of Timur Khabibulin and Aleksandr Nedovyesov.
Kukushkin stunned world number 28 Borna Coric in Friday's singles.
Sunday's remaining second reverse singles will not be played, according to the organisers.