In the first set both men traded five holds of serve before Kukushkin appeared to make the crucial breakthrough and go in front 6-5.
However, American Isner responded by breaking straight back to love in the next, forcing a tie-break.
It would be Kazakhstan's Kukushkin who won through, clinching his second set point to claim the breaker 8-6.
A tie-break would again be needed in the second set after no breaks of serve, with this time Isner needing just one mini-break to take the set.
Set three would be the only one without a tie-break as Isner crucially broke when 4-3 ahead before serving out the set 6-3 to take a 2-1 lead.
In the fourth neither man wobbled on serve, and in the breaker it was Isner who stayed cool on his first match point to reach the last 32.