Roger Federer's run at the Miami Open was one point from ending. Down 6-4 in a third-set tiebreaker to Tomas Berdych, the situation was officially dire.

Yet even in that moment, Federer still felt some hope.

"I had belief I could turn it around, even then," he said.

Somehow, he was right, and his stellar start to 2017 continued. The fourth-seeded Federer fought off those two match points and beat the 10th-seeded Berdych 6-2, 3-6, 7-6 (6) in the quarterfinals on Thursday – avenging a third-set tiebreak loss to Berdych at Key Biscayne seven years ago in a match he still thinks he should have won.

"I got incredibly lucky," Federer said. "Could have gone either way. Felt like maybe this one I should have lost."

Federer feels right at home at Key Biscayne, and so does Caroline Wozniacki – with good reason, since she sometimes practices at the facility. The 12th-seeded Wozniacki, a part-time South Florida resident, made the women's final for the first time in 10 tries by topping second-seeded Karolina Pliskova 5-7, 6-1, 6-1.

"This is one of the few tournaments where I've never made a finals," Wozniacki said. "I think my best result here was semifinals five years ago. It's always been a tournament where I wouldn't say I struggle, but I've just not had the results I wanted."

Federer improved to 17-1 this year and will face No. 12 Nick Kyrgios in the semifinals on Friday.

Kyrgios defeated 16th-seeded Alexander Zverev 6-4, 6-7 (9), 6-3 in the last of the men's quarters, a match that lasted 2 1/2 hours. Kyrgios had 16 aces, no double faults and never faced a break point, though his 19-year-old opponent saved five match points before falling.

Zverev fought off three match points in the second-set tiebreaker, and won the set when Kyrgios — who pulled off two between-the-legs shots on the same point in the first set tried another that didn't work.

"I don't know what I was thinking," Kyrgios said.

He recovered and gets to face Federer, whom he called "the greatest of all time … my favourite tennis player." Federer-Kyrgios is a rematch sort of from this year's quarterfinals at Indian Wells, a match where Kyrgios withdrew beforehand with an illness.

Rafael Nadal and Fabio Fognini are the other men's semifinalists, meaning there's still a chance for Federer-Nadal on Sunday for the men's crown.

"I would love it," Federer said.

Federer is now 4-0 in tiebreakers this year at Key Biscayne, none of the first three as pressure-packed as the one he needed in the quarters. He was serving for the match at 5-3 in the third and got broken, had a match point in the next game and couldn't convert, then was down 6-4 in the breaker before winning the final four points.

Berdych actually won 91 points to Federer's 89. He needed 92 – and after coming up with big shot after big shot in the final two sets, he wound up going out on a double-fault.

"I just lost by one point. That's what happened. Very simple, very straightforward," Berdych said. "He was the one serving out the match, didn't make it. I had a match point, didn't make it. I had two, didn't make it. So what else to say?”

Like Federer, Wozniacki rallied, albeit with far less drama. She won 12 of the last 14 games.

"I got a good start to the second set and that kind of got me fired up," Wozniacki said.

This will be the second consecutive time two double-digit seeds make the women's final at Key Biscayne, after No. 13 Victoria Azarenka beat No. 15 Svetlana Kuznetsova a year ago.

No. 11 Venus Williams and No. 10 Johanna Konta were to play for the other spot in the women's final later Thursday night.

"It's extremely special," Wozniacki said. "Having a place here, training here in the offseason, playing kind of on home advantage, it's special to be in my first finals here. I'm extremely excited."