She was finally able to answer back with a win in the sixth game to bring her to 4-2, though that also came with a language warning from the umpire.
She will play the victor of Tuesday's night match between Serena Williams and Karolina Pliskova in the last four.
Karolina Pliskova, the No. 8 seed, knocked off Williams in the 2016 semifinals before the 23-time Grand Slam champion missed last year's tournament, when she gave birth.
Next up for Stephens, who at number three is the highest seed left in the women's draw, is a repeat of last year's quarter-final against Latvian Anastasija Sevastova.
"It's always tough to play (Stephens), I think, because her ball is so heavy", said Sevastova, who had lost 2-6, 2-6 to Stephens at the Canadian Open in August.
The match was played when the temperature pushed above 32 degrees Celsius, prompting tournament officials to implement the extreme heat policy, which has been in effect for most of the tournament.
Sevastova earned her second career title - and first in seven years - at Mallorca in 2017 - finishing at 16th in the world last year after a season that included her US Open last-eight run.
On Monday, Roger Federer and Maria Sharapova crashed out in back-to-back seismic shocks under the Arthur Ashe Stadium lights.
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He finished with 77 unforced errors and 10 double faults.
Novak Djokovic is aiming to make an eleventh US Open semi final tonight.
Sharapova, the 2006 champion, suffered a second successive last-16 exit in NY, blasted off court by Carla Suarez Navarro. It's not every day you play semi-finals of a grand slam.
Cilic made a sluggish start against the 10th-seeded Goffin but had enough gas in the tank to advance. "That's the most important thing to me", she continued.
"When you don't play big points well, the match can get away from you". She received a wild card into last year's U.S. Open and became the youngest player ever to make the girls' final.
Encouraged by her home crowd, the world number three threatened a comeback by getting the match back on serve at 4-3 but was then broken again before Sevastova served out.
Gauff said she tried both those sports, but found her niche with tennis. I think now it's something really cool, something to look forward to.
Osaka will next face Lesia Tsurenko, the 36th-ranked Ukrainian who beat Czech teenager Marketa Vondrousova 6-7 (3-7), 7-5, 6-2.