But Cilic came back in the fourth set to keep himself in contention and take the match to a final and decisive fifth set that would see the pair play for over four hours.
"I definitely left it all out there today and pulled up a bit short".
The 29-year-old, who had knocked out world number two and Australian Open champion Caroline Wozniacki in the second round, hit 31 unforced errors and won just seven points off the Japanese star's serve.
It was just the latest in what's been a dominant run for Osaka.
Millman, who stunned five-time champion Roger Federer in the fourth round, kept Djokovic running - and was kept running himself - for 2 hours 49 minutes on another hot, humid night on Arthur Ashe Stadium.
Osaka will face either 14th-seeded Madison Keys or No. 30 Carla Suarez Navarro in the first major semifinal appearance for a Japanese woman since Date reached the final four at Wimbledon in 1996.
The 20-year-old trounced Lesia Tsurenko 6-1 6-1 to become the first Japanese woman since Kimiko Date at Wimbledon in 22 years to advance to the last four of a grand slam.
Kei Nishikori advanced to the U.S. Open semi-finals on Wednesday with a grinding 2-6 6-4 7-6 (5) 4-6-6-4 win over Marin Cilic, gaining a small measure of revenge on the man who crushed his grand slam dream in the 2014 final.
Kei Nishikori is headed to the US Open semifinal following a dramatic upset victory over Marin Cilic on Wednesday. I was up 4-1 and he came back.
"It really means a lot to get to the semis", said Osaka whose post-match, on-court interview endeared her further to a growing legion of fans.
Osaka, who lived in her native Japan until she was 3, started playing tennis in NY after her father saw the Williams sisters playing at the French Open in 1999 and chose to teach his daughters.
"I'm working very hard, and we are all in the team, to make sure that Naomi one day might own the court like Serena", Bajin said. "Especially after coming from injury", he said, "I think I'm enjoying this challenge".
Naomi Osaka made the semifinals with a win over Lesia Tsurenko.
He had two game points for a 5-2 lead but a double fault and a backhand wide opened the door for Cilic, who stormed through, shouting "Got it!" when he secured the break to narrow the gap to 4-3.
The Californian has dropped just a single set so far at the 2018 edition of the United States slam, to Aleksandra Krunic in the third round, and has a 2-0 edge against her upcoming Japanese opponent. I feel it is a big chance for her.
Osaka had consecutive 50-minute matches earlier in the tournament, including a 6-0, 6-0 thrashing of Aliaksandra Sasnovich in the third round.
Keys said she will look to home support on Thursday when she resumes her rivalry with Osaka.
Thomas' reaction to the lack of a resolution was clear in his post: "The disrespect has been noted and will not be forgotten". Thomas has since reported to the Seahawks, but he still doesn't appear to be thrilled with his situation.
The report added that five cars and a mini bus are trapped underneath piles of debris from the collpaed section of the bridge . My thoughts are with the families of the victims. "I pray that those who are injured recover at the earliest".
Kaia Kanepi of Estonia congratulates Serena Williams at the net after their match during the seventh day of the US Open. But having taken her first set against Williams in five career meetings, there would not be a second.
The system was lashing Florida with tropical-storm-force winds and heavy rains Monday morning, according to the hurricane center. At the time, peak sustained winds were 45 miles per hour , but wind gusts were in the 50 mph-to-60 miles per hour range.
The victor came after 62 minutes when Olivier Ntcham swept home to find the net at the end of a swift counter attack. A fuming Gerrard told Sky Sports: 'It's definitely a foul. "I have seen it again five times and it is a foul".
He controls the first touch on the half turn, swings round and shoots above Etheridge's outstretched arms. Goals form Shkodran Mustafi, Pierre-Emerick Aubameyang and Alexandre Lacazette sealed an important win.
Del Potro - the 2009 victor of the men's singles at Flushing Meadows - breezed past Borna Coric in the final match of the day. Serena, seeded 17th, routed Venus 6-1, 6-2 in matching the most-lopsided victory in the Williams sisters' series.
From the coach, to the physio to maybe even a counsellor, these are the people you hire to get the best out of you. De Minaur took the opening two sets, including a bagel in the second, before Tiafoe fought back to make it 2-1.
Storm Gordon weakens after landfall in southern US
A child has been killed in Florida after a tree was blown on top of his mobile home in an incident which may be linked to Gordon. Brandon Drake, a meteorologist for the National Weather Service in Topeka, said theories differ on how the storm will progress.
Amazon Hits $1 Trillion Valuation
Amazon has become the second publicly traded company to be worth US$1 trillion (NZ$1.52 trillion), hot on the heels of Apple . Amazon has briefly become the second US-listed firm to have a market value of more than $1 trillion (£779bn).
John Millman Just Knocked Off Roger Federer At The US Open
The 13-times grand slam victor had made easy work of the opening set, breaking Sousa twice, the second time to close out the set. Kei Nishikori , runner-up in 2014, was also pleased to get through in three sets in blazing sunshine on Louis Armstrong Stadium.
Zverev sails into third round at U.S. Open
That was the first career Grand Slam final for Keys, who was one of four American women in the semifinals a year ago at the U.S. Former world No. 1 Novak Djokovic will be back on Arthur Ashe in prime time, facing American Tennys Sandgren.
Energy department intervenes to keep petrol price increase down
Radebe said that the fuel price was driven by global factors - such as the rand exchange rate, the worldwide price of crude oil. If the current trend continues, both petrol and diesel prices will have a double digit price growth in the current fiscal year.
Colin Kaepernick appears in provocative Nike ad
The initial image, announced through Rovell, has a close up of Kaepernick's face with the caption: "Believe in something . Colin Kaepernick is not on an National Football League roster, but he does have his own ad campaign, courtesy of Nike .