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Dimitrov went into the contest having won their only previous meeting on a hard court, but Kukushkin started the match strongly, breaking the Bulgarian three times on the way to taking the opening set.
Kukushkin immediately dropped set at the start of the second as Dimitrov coasted into an early advantage, but the Kazakh's aggressive play saw him get back on level terms before he dominated a tie-break to move to the brink of a surprise win.
However, Dimitrov showed his class to ease through the third set to keep his hopes alive, and a single break of serve in the fourth was enough to ensure that the match went into a decider.
Kukushkin rediscovered his rhythm in the final set though, breaking the world number 17 before completing a shock victory at Flushing Meadows.