Cilic won the first set 6-2 but Nishikori fought back to win the next two 6-4 7-6.
But Cilic, US Open champion in 2014 refused to back down and took the match into a decider by winning the fourth set 6-4.
Nishikori got the early break in the fifth set and led by 4-1 but Cilic got back on serve at 4-5.
However the Croatian faltered in his final service game, slipping to 15-40 and Nishikori won the match with a cross-court forehand return.
It means Nishikori reaches a Grand Slam semi-final for the third time in his career.
And it is also the first time a Japanese man and woman have made the semi-finals of the same Grand Slam after Naomi Osaka won her quarter-final earlier today.
Nishikori said: “It was really tough. I was up 4-1 and he came back.
“We always play full sets. It was a great battle with Marin.
“I wish I don’t go to five sets every time.
“I tried to fight for every point even though I was really tired.
“At the end, I focused again – focused on one point at a time.”