The second seed had lost the first set in each of his two opening matches and that trend continued on Saturday against Kukushkin, the world number 63.
However, Nadal had little trouble thereafter, showing his class to prevail 6-7 (4-7) 6-1 6-1 6-1 under the Centre Court roof on a day when play elsewhere at the All England Club was delayed due to rain.
Kukushkin had failed to take a set off Nadal in their two previous meetings, but soon changed that with the aid of some ferocious forehands and excellent movement.
Two-time Wimbledon champion Nadal hit back in impressive fashion, breaking twice in the second set to draw level as his opponent's level of intensity dropped.
Kukushkin continued to find sporadic forehand winners yet the 26-year-old slowly saw his game unravel as Nadal comfortably saw out the match to set up a fourth-round clash with Nick Kyrgios, who triumphed 3-6 6-3 7-5 6-2 over Jiri Vesely.
Roger Federer barely broke sweat as he convincingly swept past Santiago Giraldo and into the next round.
The seven-time winner was at his supreme best as he recorded a seventh consecutive win on grass with a 6-3 6-1 6-3 win on Saturday.
Federer only needed 82 minutes to complete victory and continue his record of not dropping a set in this year's tournament, while Giraldo's search for a second win over a top-10 player continues.
The Swiss maestro will now face Tommy Robredo in the fourth round after the Spaniard recorded a 6-2 6-4 6-7 (5-7) 4-6 6-3 win over number 15 seed Jerzy Janowicz.
That game reached its conclusion after most matches had called it a night due to bad light.
Simone Bolelli and Kei Nishikori were not so lucky, however, and will return to the court on Monday tied at 3-3 in the deciding set.
The winner of that clash will meet eighth seed Milos Raonic in the next round - the Canadian a 7-6 (7-2) 7-6 (7-4) 6-2 winner over Lukasz Kubot.