The Spaniard has made uncharacteristically slow starts in every match at this year's tournament to lose the first set in each, and he did the same on Saturday - only for the two-time champion to come through for the win.
The second seed took time to get going under the roof on Centre Court as his Kazakh opponent claimed the opener on a tie-break, but Nadal was not to be denied as he recorded a 6-7 (4-7) 6-1 6-1 6-1 victory.
Nadal roared back into the match with his trademark energetic style, and by winning 18 of the final 21 games he booked a fourth-round place for the first time since 2011.
"I finished all my matches a better player than when started and that is very positive," he told BBC Sport.
"I'm very happy to be in the second week after two years losing in the first and second rounds. I am fighting for every ball and playing to find right solutions in every game.
"He (Kukushkin) played very aggressive and I made a few mistakes with his second serve. I made some unforced errors. That took us to a tie-break (in the first set) and in that I didn't serve my best."
With rain affecting the outside courts on Saturday, Nadal will have to wait to find out his opponent after Jiri Vesely and Nick Kyrgios were only able to play six games of their third-round match.
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