"I'm happy today to have come through somehow", Federer said.
Federer next will play the victor of Thursday s later quarter-final between two of the ATP s up and coming stars, Australia s Nick Kyrgios and Germany s Alexander Zverev.
A day after saving three match points and beating John Isner, the 19-year-old Zverev dominated the final two sets against Wawrinka, breaking twice in the second and three times in the third to earn a quarter-final encounter with either Belgian David Goffin or Australian Nick Kyrgios.
Bautista had managed to break Federer's serve on one previous occasion, at the US Open in 2014, his first appearance on the pro circuit, but after that he has found the Swiss dynamo virtually impossible to break, yet he managed to do so once again at Crandon Park on Tuesday.
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"I enjoy winning breakers because those are the ones that are going to make you win a tournament or not sometimes", he said.
It was the second tough loss this month for Berdych, who lost at Indian Wells after leading 6-1, 5-2 against Japan's Yoshihito Nishioka.
"I thought Roberto did a good job of making shots, staying offensive when he could, fending off a lot of good shots that I hit", Federer said. A double fault from Berdych at 6/7 wrapped up the match after one hour and 57 minutes.
Federer hadn't saved a match point en route to victory since erasing three in coming back from two sets down to defeat Marin Cilic in the 2016 Wimbledon quarter-finals.
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"I guess I like to see that [Berdych] went out and went for it and kept believing until the very end", Federer said of his opponent's final two serves. The Czech 10th seed was a 6-3, 7-5 victor over France's Adrian Mannarino. That's what I've always tried to explain to people, that I do feel that I have improved.
Venus Williams moved a step closer to her first Key Biscayne title since 2001 when she beat top-ranked Angelique Kerber 7-5, 6-3 Wednesday night in the quarterfinals of the Miami Open.
Pliskova ended the hopes of Croat veteran Mirjana Lucic-Baroni with a 6-3, 6-4 win while Wozniacki made sure there would be no all-Czech semi-final by dispatching Lucie Safarova 6-4. 6-3. "It was really humid, not easy to play some great tennis".
His 4-6, 6-2, 6-1 triumph over Wawrinka, was his second win over the Swiss, having also defeated him in the St. Petersburg final last September.
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On the men's side, Kei Nishikori rallied in the final set to beat Argentina's Federico Delbonis and reach the quarterfinals.