After world No.20 Zverev refused to yield in an enthralling second-set tiebreak, Kyrgios rallied to break serve in the sixth game of the third set before taking victory in two-and-a-half hours.
Flustered, he lost the game, which turned out to be the only service break of the set, and Zverev gave Murphy a piece of his mind at the next change over. He saved a break point in the second game, but Berdych was increasingly on the offensive, taking Federer's serve early, and hitting clean and penetrating ground strokes.
The No.10 seed then had a golden opportunity for his seventh win in 24 contests against Federer when he hit a backhand victor to give himself two match points at 6-4, the first one on his own serve. But first came No4 seed, two-time champion, the 35-year-old Roger Federer against a man he had played 23 times, No10 seed and former Miami finalist, 31-year-old Tomas Berdych. Federer had to fight off two match points before securing a win, setting up a semi-final showdown against Kyrgios.
"They like the big stage and the belief they have makes them tough to play in any given match", said Federer of the ATP's rising stars.
"I definitely got very lucky at the end, but I think I showed great heart today".
At 30-love down in the game, many players would get flustered at the idea of losing another point, but Kyrgios kept his cool and placed his shot perfectly across court.
"It's obviously not something that you want to finish the tournament like", he said.
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Roger Federer and Caroline Wozniacki rolled through the first few matches of the Miami Open, never facing any significant challenges. At one point in the Zverev match, Kyrgios recommended his opponent challenge a call that went against him.
Rafael Nadal has already qualified for the semi final.
Nick Kyrgios of Australia reacts in his match against Alexander Zverev of Germanyin Florida.
"He's my favourite tennis player, so I'm going to enjoy that moment".
Two ridiculous backhand lobs - one which left Kyrgios stranded and another which forced him to play a tweener into the net - allowed the world No.20 to level the match at a set apiece. We're going to play a lot more times and he's only going to get better.
"He's playing incredible at the moment", Kyrgios said of the veteran after his quarter-final win on Thursday.
Fognini has become a nemesis for Nadal, splitting the six most recent meetings between the two, including a come-from-behind win at the 2015 U.S. Open. As is well known, Nadal has been going through the roughest part of his career, both in terms of major-tournament success and Masters 1000 prosperity.