Rune, the number six seed, came back from the brink to win four straight games and end the Japanese veteran's challenge in front of a lively crowd in Tokyo.
"It was about getting that first serve, and if I could save that match point I knew that I could put on pressure," said Rune, who will play either defending champion Ben Shelton or France's Arthur Fils in the semi-finals.
"The momentum shifted a little bit. I lifted myself up, I was more positive."
Nishikori has been plagued by injuries and he returned to Grand Slam competition at the French Open in May after an absence of almost three years.
He has been in inspired form this week in Tokyo and took control of the match early against Rune.
"It was a great week for me - great three matches with tough opponents," said Nishikori, who reached the US Open final a decade ago and was once ranked number four in the world.
"I would say this was the best week so far. I played very solid, I raised my level."
Rune needed a medical time-out in the third set and he said that "in one moment I thought it was not going to be".
"But I just kept my belief and trusted my fighting spirit, that was really all I could do and I managed to get through," he said.
"Definitely one for the books."
Humbert advances
France's Ugo Humbert also advanced to the semi-finals after his British opponent Jack Draper retired early in the second set.
Draper earlier this month became the first British man to reach the US Open semi-finals since Andy Murray won the title 12 years ago.
He was looking to build on his success in Tokyo but he needed medical treatment on his upper body late in a first set that Humbert won 7-5.
The Frenchman was leading 2-1 in the second when Draper signalled that he could not continue.
"I feel sorry for him because he was not 100 per cent but at least we played an amazing first set with a great level, great intensity," said Humbert.
"I'm just happy to be in the semis."
Draper had not dropped serve in two matches heading into the quarter-finals in Tokyo but Humbert broke him three times in a hard-fought first set.
Humbert, who is ranked 19th in the world and has won two titles on the ATP tour this year, will face the Czech Republic's Tomas Machac in the semi-finals.
Machac beat American qualifier Alex Michelsen 7-6(2), 6-3 in the day's first quarter-final.
"Everyone is playing so good, the ranking doesn't matter," said Humbert.
"Everyone plays really good, everyone can beat everyone. I just try to take my chance, to play my game, and let's see what happens."