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The Briton, seeded second, won 7-5 6-1 6-4 on Centre Court and will next face French 12th seed Jo-Wilfried Tsonga on Wednesday.
After matching Murray for 40 minutes, Kyrgios' challenge faded quickly under sustained pressure from the Scot.
Murray, the 2013 champion, is into the last eight without dropping a set.
The 29-year-old has now won eight matches in a row on grass, and joins Pete Sampras and John McEnroe in reaching nine consecutive Wimbledon quarter-finals.
Murray had beaten Kyrgios at the other three Grand Slams - and he brushed aside any thought this might be the Australian's time with a brilliant display.
The world number two was focused throughout, winning 80% of points on the Kyrgios second serve and denying the 21-year-old a single break point.
"The first set was very tight. I managed to get the break at 6-5 but it was tight up to that point," Murray told BBC Sport.
"The second set was much more comfortable, Nick lost his focus a bit. I was able to dictate a lot of the rallies."
Asked about the growing expectation of a second title following top seed Novak Djokovic's defeat on Saturday, Murray added: "Every year I play here it's the same.
"I try my best to win my matches and go as deep as possible.
"The matches, as you progress, get tougher. Tsonga is next who is a tough grass-court player and I'll have to play very well to win that one."
Kyrgios said: "As soon as I lost the first set, I just lost belief. It obviously felt like a mountain to climb after losing the first.
"He played pretty well, as well. I don't think he missed too many balls either."
It is a measure of how sharp Murray is looking that he won the opening set despite Kyrgios making 30 of 34 first serves - the bedrock of his game.
There were flashes of brilliance all over the court from the Australian and a tie-break appeared inevitable after 11 games without a break point, but Murray grabbed the initiative with a brilliant return game.
A backhand pass gave the Scot a 0-30 lead before a clever cross-court chip drew a loose volley from Kyrgios, who suddenly found himself staring at three set points.
Two were saved with big serves but Murray drilled a cross-court forehand on the third that Kyrgios could only divert into the net.
If the first set was a tight, tense contest, the second disappeared in just 26 minutes as Murray maintained the quality of a tournament favourite, while Kyrgios' belief appeared to drain away.
Murray was now peppering the baseline with his returns, drawing Kyrgios into rallies, and he made the breakthrough with another pass that the Australian could only net.
Two games later and the pair's much-discussed friendship appeared briefly on hold as Murray slowed the quick-fire Kyrgios serving pace, prompting the Australian to thrash a wild forehand out and hand over another break.
Murray crunched down his sixth ace to clinch a two-set lead and broke again early in the third set thanks to three thumping forehands.
By now there was little prospect of the close contest many had predicted, and there were murmurs from the crowd at some careless play from the Australian, but nothing was going to distract Murray.
John McEnroe, three-time Wimbledon champion: "Murray took Kyrgios completely out of the match. He did not look under any stress at all. Considering what he was up against in Kyrgios, it was very impressive. The outcome was never in doubt.
"Kyrgios went through the motions in the second set. The truth is I don't think he would have won anyway, even if he played his best. The level Murray played at was extremely high. Everyone respects him - and the level he has reached - more than they ever have."
Pat Cash, 1987 Wimbledon champion: "That was a real test for Murray. Nick served nearly 90% in the first set and still didn't get to a tie-break. Sometimes you think about Nick and think he needs some rewiring. I don't think he is trying sometimes, there's no doubt about it, but that's the way he plays.
"People are watching that match and think what's going on? They might come away from it feeling a bit short-changed."
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Andy Murray saw off the threat of Australian rising star Nick Kyrgios with a clinical performance to reach his ninth Wimbledon quarter-final.