Kukushkin beat Gael Monfils in five gruelling sets on Saturday and those exertions seemed to hinder him in the scorching Melbourne heat.
Three unforced errors in the first game handed Murray a break and the Scot struck again for a 3-0 lead.
Kukushkin finally displayed some fight to hit back in game four, only to relinquish his serve again after pushing two backhands wide.
Jonathan OverendBBC tennis correspondent
“This was entirely predictable. Kukushkin, having played back-to-back five setters, looked off the pace from the start. It was going to be a thrashing or a retirement and Kukushkin chose the second option, having received treatment for a hip flexor injury. Murray looked sharp, as he has all tournament, and things are building nicely. What a bonus to only have to play for 49 minutes on the hottest day of the tournament so far. After the fun match against Llodra, now the half-day against Kukushkin, this could be the break he needs to make all the difference in the second week”
The world number 92 gave an indication he was not 100% fit when he called for the trainer at the next change of ends, and Murray broke once more to wrap up the 22-minute first set.
A lapse in focus saw the world number four relinquish an early break in the second set with a poor backhand into the net.
But a struggling Kukushkin, who started rushing the net in a bid to shorten points, handed the momentum straight back.
Murray breached his defences with some clean hitting to go 5-1 up before serving out another 22-minute set to love.
Having required more treatment and then lost his serve for an eighth successive time, Kukushkin called it quits.
Next up for Murray is Nishikori
who stunned sixth seed and 2008 runner-up Tsonga 2-6 6-2 6-1 3-6 6-3 in three hours and 30 minutes.
"I'm feeling unbelievable," Nishikori said. "To reach my first quarter-final and beating Tsonga makes me really happy.
"Hopefully it's big in Japan. A lot of people messaged me when I got to last 16. Now I'm in the quarter-finals, it's really exciting."
In the mixed doubles,
Britain's Colin Fleming
reached the quarter-finals with his partner, the South African-born American Liezel Huber.
The duo defeated American Steve Lipsky and Vladimira Uhlirova of the Czech Republic 6-3 6-2.
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