She was pushed all the way here by Kvitova in a captivating duel, eventually prevailing 6-3 3-6 7-6 (7/2) in a deciding tie-break after two hours and 35 minutes.

Williams said: "It definitely felt like a special match. No easy moments, not easy to hold serve or break serve. This match meant a lot to me, obviously, playing at home and of course it being a major.

"It means a lot to her, coming back and being able to compete in this major and to prove obviously to herself that she could defeat anything no matter what's thrown at her. It was amazing to see her shine today.

"Tie-breakers, you have to play smart but you have to be aggressive. You can't just sit back and hope. I didn't want to hope. I wanted to be doing something about my future."

Williams' victory also ensures she will return to the top five in the rankings for the first time in six years, irrespective of the result of her next match against fellow countrywoman Sloane Stephens.

The pair completed half of what could be an American clean sweep of the semi-finals, with Coco Vandeweghe and Madison Keys playing their matches on Wednesday.

"It's been a great two weeks for American tennis," said Williams. "Seeing all the American players in the draw and all of them advancing so deep and competing so well.

"All I have known all my life was great American players. So it's great to see this resurgence, and I hope it can continue."

For Kvitova, there were mixed emotions. It has been a remarkable fortnight for the Czech as she continues to recover physically and mentally from the knife attack that nearly cost her her career and might have cost her so much more.

She said: "It wasn't enough. But it was pretty close, pretty intense. It was just about a few points. She played very well in the tie-break.

"I'm glad that I am still able to compete on the high level against the top players, which I hope will give me some confidence for hard work again to continue to play well, to know that it's still there somewhere.

"I'm not sure about (winning) titles of the grand slam. Of course that's why I'm playing tennis, and that's why I wanted to be back and playing, and it is a big motivation.

"But I'm still on the earth, and I know it's still a lot of work. It was a quarter-final, but I know how tough it is to get a title. It was pretty close but pretty far, as well."

This has been a tournament full of inspirational stories and Stephens' is another after reaching her first US Open semi-final only two months into her comeback from a foot injury.

The 24-year-old was unable to walk after undergoing surgery in January but has hit a remarkable hot streak, making the last four at three successive tournaments.

Like Williams, Stephens had to do it the hard way, coming from 3-1 down in the final set to defeat Maria Sharapova's conqueror Anastasija Sevastova 6-3 3-6 7-6 (7/4).

"It's incredible, amazing," she said. "If someone would have told me when I started at Wimbledon that I'd be in the semi-finals or making three semifinals back-to-back, I would have said they're crazy."

Stephens was hailed as the natural successor to the Williams sisters when she made her first grand slam semi-final in Australia in 2013, but had struggled to back it up until now.

She has a different attitude to tennis after being forced to take a break and believes she is ready to play on the big stage.

Stephens said: "The first one was quite overwhelming. But this one feels good. I think I have a better perspective and just looking at the game totally different. I'm just happy with where I'm at and looking to build on it."

And playing Williams will make the occasion extra special.

"I think Venus is just our leader," said Stephens. "She's a great player, a great person. Being on Fed Cup teams with her, there is not anything bad you can say about Venus.

"I'm just honoured to be able to play at the same time as her. I'm happy she's still playing. She means a lot to the game."

AP