Kukushkin, ranked 63rd in the world, held his own during the opening set as some clean hitting from the baseline had Nadal stretched.
Both players continued to hold serve, but it was Kukushkin who came through the tie-break 7-4 thanks to some crisp winners against the two-time champion.
However, Nadal fought back in determined fashion, crushing the Kazakh 6-1 to take the second set and level the match.
The second seed was even more dominant in the third, winning 82% of points on his first serve and 100% on his second as he took control of the contest.
Nadal broke instantly at the start of the set before two more breaks saw him come through 6-1.
It was a similar story in the fourth, with the former champion opening up a 3-0 lead before sealing the win 6-7(4) 6-1 6-1 6-1 with two hours and 36 minutes on the clock.