The Japanese 21st seed, a finalist at Flushing Meadows in 2014 and a semi-finalist in 2016, fired 29 winners to end the run of Germany's Kohlschreiber, who had beaten fourth seed Alexander Zverev in the previous round.
Nishikori appeared to be racing towards an easy win before Kohlschreiber unexpectedly broke the 28-year-old to level the third set at 5-5.
But that only delayed the inevitable as Nishikori kept his focus to seal the win and extend his perfect record against the German to 3-0.
"(I was) really lucky to finish in three sets -- it was really hot on the court," a sweat-soaked Nishikori said in an on-court interview. "I think we both struggled a little bit with the heat."
Nishikori later told reporters that the brutal conditions made it difficult to keep his head in the match.
"It's not easy with the heat. You know, I couldn't think too much between the points. It's not easy to focus (on) every point, you know. You lose some concentration," he said.
Kohlschreiber thought the heat -- as well as Nishikori's slick performance -- made the match particularly challenging.
"He pushed, especially in my weaknesses, pretty good, you know, returning my second serve very aggressively, very deep," Kohlschreiber told reporters. "He didn't make many mistakes. He pushed me around. He was too much in control of the game."
Nishikori next faces either seventh-seeded Croat Marin Cilic, which would be a rematch of the 2014 final, or Belgian 10th seed David Goffin in the last eight.
(Reporting By Amy Tennery, editing by Pritha Sarkar and Ken Ferris)
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