But the Australian jokingly played down the shot after the match despite his wide-eyed admiration on court.
The point of the match, and one of the shots of the tournament, came when Federer blocked Kyrgios' serve to the backhand and the 23-year-old Australian guided a drop shot cross-court toward the photographers' pit.
But the best from the match, a highlight that would be played over-and-over came in the third set.
Roger Federer taunted Nick Kyrgios for his shot-making audacity after delivering a tennis masterclass to send the flashy Australian packing from the US Open in NY. "It just comes out, and it's just emotions, '" said Williams, a six-time US Open champion who finished with more than twice as many winners as Kanepi, 47-22.
"I am a big fan of John [Millman] - not many people know too much about him but he is such a hardworking guy and those guys have all the respect from those of us at the top", Federer said about his next challenge.
Other than those big names, however, most of Saturday's results from the fourth and final Grand Slam event of the year were peppered with surprise. Zverev still has never made the fourth round in visits to NY after being beaten 6-7 (1), 6-4, 6-1, 6-3 by Philipp Kohlschreiber in an all-German matchup, while Joao Sousa defeated Pouille 7-6 (5), 4-6, 7-6 (4), 7-6 (5).
At Roland Garros and the US Open, he has never reached the second week.
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Federer showed no signs of his advancing years to produce a spectacular moment which left nearly everyone inside Arthur Ashe Stadium on their feet in amazement and caused a stir on social media.
The contrast between Kyrgios' mindset and Federer's was not lost on the talented, if temperamental, Australian.
Kohlschreiber will seek a quarter-final berth against 2014 runner-up Kei Nishikori after the Japanese downed Argentine Diego Schwartzman 6-4, 6-4, 5-7, 6-1.
Federer advances to play Australia's John Millman in the fourth round, and beyond that looms a potential quarterfinal matchup with sixth-seeded Novak Djokovic, who made quick work of Richard Gasquet, 6-2, 6-3, 6-3, on Saturday night. All the pressure was off him.
But he said no matter what he tried, Federer had the answers. "When he gets in front, there's not much you can do".
"There is a lot more things to explore".
Now, having lost for the third time to Federer, Kyrgios believes the great Swiss can steer him on the right path.
Cibulkova, the 29th seed, moves on to face 2017 finalist Madison Keys, who shook off a slow start to beat Serbian Aleksandra Krunic 4-6, 6-1, 6-2. On the first three, Kyrgios missed a shot.