The world number seven, who held a 4-0 record over Kazakh Kukushkin, struggled during the first set as he made a number of errors to concede an early break.
Kukushkin, ranked 46 places below the Spaniard, didn’t relinquish the early advantage and held his serve and his nerve to take the opening set.
However, Ferrer showed his experience during the second stanza to break his opponent and level the match at one set each.
With the momentum behind him, Ferrer claimed the final five games of the contest to claim victory in one hour and 56 minutes.
The Spaniard will now face either Juan Monaco or Andreas Seppi.
Earlier in the day youngster Dominic Thiem saved a match point and fought from 3-5 down in the final set to beat Marcel Granollers 2-6 6-3 7-5.
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