Murray, the third seed, was ultimately too solid for the 20-year-old Kyrgios who played some scintillating shots but fired too many errors to really trouble the Scot.
Kyrgios, the 29th seed, said later he had been troubled by an elbow injury that prevented him serving at full power.
"I don't know what the problem is, but you guys can see for yourselves on the stats I'm not serving anywhere near the pace I usually serve. It's heartbreaking, really," Kyrgios, whose language was as colourful as his pink and blue kit, said.
Aged 28, two-times grand slam champion Murray can probably relate to the emotional rollercoaster that faces the new generation, but these days his level rarely dips.
He was initially troubled by the pace and unpredictability of Kyrgios's groundstrokes though, and the opening games were littered with errors from both players.
Kyrgios served two double faults to drop serve at 1-1 but was then gifted a break of his own when Murray also served a double, berating himself loudly for his carelessness.
Murray saved a break point at 2-3 with a forehand winner and the expected fireworks fizzed briefly in the following game with Kyrgios producing a miraculous between-the-legs lob, only to drop serve again with a rash forehand.
The Scot sealed the opening set with an ace and got the first break of serve in the second, at 2-2, when he cleaned the line with a clinical backhand winner.
A forehand rocket earned Kyrgios two break points in the following game but he missed the first with the court gaping and once Murray held, the writing was on the wall for the Australian who had hoped to become the first man from Down Under to reach the last 16 here since Lleyton Hewitt in 2007.
Murray claimed victory with a stunning backhand winner to take his winning streak on clay to 13 and will play Frenchman Jeremy Chardy for a place in the quarter-finals.
"Sometimes you don't feel like you're able to control what's happening out there because he's hitting huge shots and sometimes playing shots that no one else tries," Murray said.
"You just have to be kind of on your toes at all times and just try to be ready for something different.
"I think obviously today he wasn't serving as big as usual, so wasn't getting as many free points."
(Reporting by Martyn Herman; editing by Justin Palmer)
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