The British number one led Kukushkin 6-1 6-1 1-0 when the 24-year-old was forced to retire.
Talking to reporters after his match, Murray insisted that he will keep himself fresh as a result of his short encounter.
"It's perfect because you conserve energy," said the world number four. "You just need to make sure, you know, today and tomorrow you hit enough balls to make sure you don't lose any rhythm.
"There was no rallies out there, I need to make sure I move around a little bit so I don't stiffen up in any way.
"But you can't look at it as a negative. At this stage of the tournament to be off the court in 45 minutes or so isn't bad."
Murray will play Japan's Kei Nishikori in the quarter-final.