Under blazing sun Murray, seeking Britain's first grand slam title since before World War II, was always in command and he led 6-1, 6-1, 1-0 when Kukushkin pulled out with a hip muscle injury on 49 minutes.
The truncated match ended an entertaining run by Kukushkin, inevitably dubbed "Borat" by the British press, who won an oddball encounter in the previous round to become Kazakhstan's best male grand slam performer.
"It's so hot on the court," said Murray, who will play Jo-Wilfried Tsonga or Kei Nishikori in the quarter-finals.
"It's obviously good for me that I got to conserve a bit of energy. It's tough for him — it's his first fourth round in a slam, and he's obviously struggling."
Murray remains on course for a semi-final with defending champion Novak Djokovic, who plays Australian veteran Lleyton Hewitt later, while Rafael Nadal and Roger Federer are into the last eight on the other side of the draw.
In the women's competition, Kvitova reached the quarter-finals with a 6-2, 7-6 (7/2) win over Ana Ivanovic — but not before a temporary meltdown in which she embarrassingly swung and missed a simple overhead.
The cringe-making mistake late in the second set precipitated a brief nosedive and a flurry of errors but tough Kvitova steadied herself to seize control of the all-important tiebreak.
"It was quite a hard match at the end of the second set," she said, adding that she was happy to be off the court early on a typically hot Melbourne day.
"I think I was quite lucky I played my match at 11 o'clock. It's hot, but we are in Australia so we are ready for it."
Kvitova will play either Sara Errani or China's Zheng Jie in the next round, while five-time champion Serena Williams takes on Ekaterina Makarova later and Maria Sharapova, the 2008 winner, faces Sabine Lisicki.
Kukushkin had caused bemusement in his five-set win over Gael Monfils, who was severely restricted by a back injury. But the Kazakh was unable to press his advantage, resulting in a series of rallies played at half-pace.
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