Djokovic, for whom this was an 11th consecutive quarter-final victory, has called on US Open organisers to find a solution.
The No. 21 seed continued his strong season after returning from a wrist injury that forced him to miss the U.S. Open past year and will play No. 6 seed Novak Djokovic on Friday.
The victory for Nishikori means that for the first time in the open era there will be a Japanese man and woman contesting the semifinals of a grand slam as Naomi Osaka cruised past Ukraine's Lesia Tsurenko 6-1 6-1.
The popular Queenslander also said he copped online abuse after his first-round loss to Taro Daniel at Winstom Salem the week before the US Open.
Rather than listening to a player or manager trotting out excuses, you are hearing the genuine concerns of a competitor over a proper issue that needs fixing and demands to be dealt with.
Osaka, 20, came back from a break down in the final set to beat red-hot Aryna Sabalenka of Belarus in the fourth round while 36th-ranked Tsurenko - who beat world number two Caroline Wozniacki in the second round - edged Czech teenager Marketa Vondrousova.
Even in the final set Nishikori looked comfortable with a 4-1 lead before Cilic levelled, only for the Croat to lose the final two games.
Osaka had it much easier, continuing what's been a largely dominant run through the draw by winning in just 57 minutes, the third time in her five matches she didn't even have to play an hour.More news: Trump doubles down on 'treason' accusation after New York Times op-ed
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Keys, who at 14 is the highest seed left in the women's draw, has shown a determination to go one step further than a year ago, having only dropped one set through her four matches.
"I wish I don't go to five sets every time", Nishikori said. The unseeded Australian knocked out 20-time Grand Slam champion Roger Federer in the fourth round to reach his first major quarterfinal. Williams has won further 73 career titles so far in her career that has spanned over two decades. "I could use the break, " Djokovic told Millman.
Next up is a mouth-watering clash against Argentine third seed Juan Martin del Potro and should Djokovic bag a third NY crown, he would join 14-times Grand Slam victor Pete Sampras in third on the all-time list, trailing only Roger Federer (20) and Rafael Nadal (17). Djokovic said the conditions at this year's tournament were the toughest he had known in the last 10 years.
"It's just we have seen it all tournament long - the girls and the guys out there, just sweating a lot".
Djokovic was 1/13 on break points when Millman double-faulted on a 14th opportunity at 4-4, volleying the ball away with his foot in annoyance at giving his opponent what he had been unable to take for himself. "But like I said on court, I would back myself playing during the daytime also".
"I can only talk about myself", Djokovic said.
"I try to fight every point", said Nishikori, who boasts an impressive record in decisive sets.
Nishikori stunned then-world No 1 Djokovic in the semi-finals in 2014 to become the first man from an Asian country to reach the championship match of a major.