Although not completely at his best, Murray was in total control and leading 6-1 6-1 1-0 when the Kazakh decided enough was enough.
Kukushkin, the world No.92, had called for the trainer in the first set - with his exertions from a five-set epics against Gael Monfils in his previous match appearing to take their toll.
And in oppressive heat, Kukushkin's physical state appeared to get worse as the match went on before he threw in the towel.
Frustratingly for Murray, though, will have been the fact that the two games he did lose in the match came on his serve.
The British No.1 had broken twice in the opening set to lead 3-0 before Kukushkin enjoyed a rare moment of success to pull one game back - with Murray sending two backhands wide.
The joy was shortlived for Kukushkin as Murray rattled off three games on the bounce to take the first set in just 22 minutes.
The one-sided nature of the contest continued in the second set as Murray broke again to go 2-0 up.
However Kukushkin punished the Murray serve to give himself hope before the Scot turned up the heat, winning the next four games to take the set.
And when Murray broke the Kazakh again at the start of third, Kukushkin decided to shake hands on the contest.
"I played him a few weeks ago (in Brisbane) and it was a tough match," Murray said. "I expected another one today but it's so hot.
"I get to conserve a bit of energy but he was obviously struggling.
"When I saw him struggling I was just trying to get a few serves in and get him running."