The 2008 and 2010 champion dropped the opening set for the third time in three matches before turning things around in a 6-7 (4/7), 6-1, 6-1, 6-1 victory.
The second seeded Spaniard blitzed world number 63 Kukushkin, who had never won a match at Wimbledon before this year and had only ever beaten one top 10 player.
Nadal, the 28-year-old winner of 14 Grand Slam titles, won 17 of the last 19 games and goes on to face either Australia's rising star Nick Kyrgios or Jiri Vesely of the Czech Republic for a place in the quarter-finals.
Nadal had dropped the first set to Martin Klizan and Lukas Rosol in his first two rounds and Kukushkin, playing in his 21st tournament of the year, employed the same free-swinging tactics to bludgeon his way through the opener.
But in front of a Royal Box containing such sporting glitterati as retired Indian cricket master Sachin Tendulkar, ex-England football captain David Beckham and former Tour de France winner Bradley Wiggins, Nadal regained his composure to race away with the win.
Kukushkin, a rare example of a player coached by his own wife, stopped the rot by getting on the board at 1-3.
But it was a brief respite as Nadal stormed to victory with 41 winners and just 12 unforced errors.
"At the beginning he was playing pretty long, no mistakes, very aggressive and I made few unforced errors with that second serve," said Nadal who has reached the second week for the first time since 2011.
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