Australian 15th seed Kyrgios, 21, is often a controversial figure, and many of his critics will again accuse him of failing to give maximum effort after pushing the tournament favourite hard in the first set before appearing to collapse mentally against the world number two.
His game faded away badly in the second two sets with the 2013 champion Murray securing a quarter-final meeting with Jo-Wilfried Tsonga on Wednesday after the Frenchman progressed when countryman Richard Gasquet was forced to retire with a back injury while trailing 4-2 in the first set on Court One.
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BBC and Eurosport pundit John McEnroe was scathing in his criticism of Kyrgios' effort during the match, accusing Murray's fourth-round opponent of being "mentally lazy" as the Scotsman progressed in only one hour and 42 minutes.
"The first set was very tight, I managed to get the break at 6-5," said Murray.
"I try my best to win my matches and go deep as possible, but the matches get tougher as you progress.
Australia's Nick Kyrgios after falling during his match against Great Britain's Andy Murray.
Image credit: Eurosport
"Tsonga is a top grass-court player, and I will need to play well to get past him.
"Every match is different. I need to keep up my own form, and if I can stay as solid and as focused as I was today, I will have a chance."
The set and the match turned in the 12th game, when a combination of Kyrgios errors and two inspired Murray backhands presented the Scot with three break points. Kyrgios saved the first two with booming serves but Murray converted the third.
Thereafter Kyrgios went walkabout, the Briton breaking him twice in a second set that flew by in 26 minutes. He broke once more in the third, closing out the contest with an ace on his third match point.
The Australian, who described his performance after the first set as "pretty pathetic", has now -- in common with a multitude of Britons -- hitched his wagon to the Murray camp.
"I hope (Murray wins)..I hope so, definitely. I think he's definitely got a great chance," Kyrgios told reporters.
Results from the Wimbledon Men's Singles Round 4 matches on Monday
2-Andy Murray (GB) beat 15-Nick Kyrgios (Australia) 7-5 6-1 6-4
28-Sam Querrey (U.S.) beat Nicolas Mahut (France) 6-4 7-6(5) 6-4
3-Roger Federer (Switzerland) beat Steve Johnson (U.S.) 6-2 6-3 7-5
9-Marin Cilic (Croatia) beat 5-Kei Nishikori (Japan) 6-1 5-1 (Nishikori retired)
32-Lucas Pouille (France) beat 19-Bernard Tomic (Australia) 6-4 4-6 3-6 6-4 10-8
12-Jo-Wilfried Tsonga (France) beat 7-Richard Gasquet (France) 4-2 (Gasquet retired)
6-Milos Raonic (Canada) beat 11-David Goffin (Belgium) 4-6 3-6 6-4 6-4 6-4
10-Tomas Berdych (Cze) v Jiri Vesely (Cze) 4-6 6-3, 7-6 (8) 6-7 (9) (Play suspended for bad light)
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