In the second set, Edmund found his game, put himself under pressure and argued with the referee - he forced the set to a tie-break where Čilić was better again.
After almost four hours on court, the World No.1 succumbed to pain high in his right leg before conceding the match to Marin Cilic 3-6, 6-3, 6-7 (7-5), 6-2, 2-0 on Tuesday night.
He said the umpire's decision to warn Cilic on break-point down was hard to understand when Seven statisticians recorded that Cilic went over the 25 second mark 40 per cent of the time, while Nadal was responsible for going over his allotted time 45 per cent of the time - and escaped any warning.
"It was an unbelievable performance from both of us and it's really unfortunate for Rafa", the mild-mannered Cilic told Jim Courier during his post-match interview.
Murray always admired Edmund's worth ethic, even inviting him to train on a number of offseason camps. "It really went well for me, because I stepped up and took responsibility".
He said: "He's got such a big game, he's 23, he's getting better".
Cilic and Edmund have played once before, with the Croatian winning in straight sets in Shanghai last October.
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Edmund was playing in the last eight of a major for the first time but showed few nerves in a 6-4 3-6 6-3 6-4 win.
"I don't know if we keep playing in this very, very hard surfaces what's going to happen in the future with our lives". "Obviously when playing in the semifinals when there is a lot at stake and not losing a serve during the match is obviously big".
By advancing to the finals, Čilić will achieve the best ranking of his career.
The first racket of the world Rafael Nadal has spent nearly four hours to beat the abuser Alexander Dolgopolov Diego Schwartzman (6:3, 6:7, 6:3, 6:3).
It was a confident enough start with a hold at the start of the third set for Edmund, but he went down a break before the first change of ends, and from then on Cilic was pushing, having anther break point chance but not converting, before breaking once more at the end of the set, and serving our comfortably at the end.
Cilic stepped up his returning game to strike the first blow for a 4-2 lead and though he appeared to be suffering with a slight leg problem, he wrapped up the opener. The victor will also be No. 1 when the next rankings are released.
Earlier, Elise Mertens stunned fourth seed Elina Svitolina of Ukraine 6-4, 6-0 to reach the semi-finals.