Murray, who had toiled against the Kazakh in the Brisbane warm-up tournament, was leading 6-1 6-1 1-0 when Kukushkin said he could not continue due to a hip flexor injury, allowing the Briton to retreat from the scorching heat on Rod Laver Arena.
World No 4 Murray, the runner-up at Melbourne Park for the last two years, will next play Jo-Wilfried Tsonga of France or Japan's Kei Nishikori.
Murray paid tribute to his new coach Ivan Lendl who watched from the stands, saying the eight-time grand slam champion had given him a welcome shot of belief in the leadup to the year's first grand slam.
"The mental side is huge in tennis, something that I've tried to get better at over the last few years ... I feel more comfortable this year," Murray said in a courtside interview.
"I don't know if you know but in the UK we haven't won a grand slam in 100 years," he quipped with interviewer and former world No 1 Jim Courier.
"It's something that I've just got to deal with, I've got better at it in the last few years."
