But the British number one battled back to break Williams on three occasions on her way to sealing victory shortly after midnight local time.
With the game finely poised at 4-4 in the second set, Konta showed an incredible burst of speed to reach a short Williams ball and win the point with a deft shot. She is yet to drop a set and the biggest test was in a momentum-shifting first set with Garbiñe Muguruza, storming through the tie-break before the Spaniard retired with illness.
Quarterfinal results: Men: Rafael Nadal bt Jack Sock 6-2, 6-3; Fabio Fognini bt Kei Nishikori 6-4, 6-2.
"The crowd, I thought it was actually very enjoyable", Federer said.
"It wasn't easy and I was just focused on my game".
"I think it makes tennis very special".
The one that became a model for young female players the world over back about a decade ago. "It's a huge part of my game", Wozniacki, a sub-3:30 marathon runner, said in a courtside interview. "So, of course, it's awesome to have him there [in the front row of the stands]", gushed Venus Williams in her post-match on-court interaction with Brad Gilbert.
Her post-match comments were more positive than her negative remarks to coach Darren Cahill after the second set.
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With the local crowd behind her, Williams, a three-time victor at Miami clearly believed she could topple the German but the first set saw both players struggle to hold their serves. "Every time we've played I have been able to raise my level", she said on Thursday night, in a tone of slight surprise.
The German will remain No. 1 next week despite her loss. But to see the happiness and the pride and the excitement, I could win the match just for that moment'.
Berdych served for the match at 6-4 in the tiebreak, but Federer reeled off four points in a row, including two unplayable serves, and secured the victory when the Czech 10th seed crumbled with a awful second serve.
"I'm really happy to have come through and am looking forward to the next round", added Konta.
"I'm going to have a tough battle on my hands", said Konta, who is assured of returning to the top 10 in the world rankings regardless of Saturday's result.
But the 36-year-old salvaged one of those breaks and made the Briton serve it out, which she did. "I can say that I'm better than before, and I'm working on it".
Konta, a late-blooming, 25-year-old from the Great Britain, was ranked outside the top 20 at this time last season but diligently worked her way into the elite top 10 and looks like she might stay awhile.