In the second set, 2017 Wimbledon champion Muguruza seemed as though she could not recover from the tough tiebreak.
Kvitova, on the comeback trail after sustaining a career-threatening injury during a home invasion late a year ago, hit back to force a third set but was unable to match her opponent's composure in the decisive tiebreak.
Never has there been a greater opportunity for one of these players to make a first Grand Slam final.
"She's a champion. I think she can win it - I hope so".
Stephens made her comeback from a foot injury only at Wimbledon and was ranked world No 957 at the end of July.
Sloane Stephens is convinced that her run to the U.S. Open semifinals is not proof that she is back to being as good a player as she was before foot surgery took her off the tour for almost a year.
Here, too, were emotional storylines of triumph over adversity. It's so wonderful to see her back and playing incredible.
She's already nearly all the way back to where she was before a foot injury suffered last summer required surgery in January. En route to the last eight, she made no mistakes despite facing some unsafe opponents like Jelena Jankovic, Alize Cornet, Caroline Garcia or Garbine Muguruza.
It's all insane uncommon, and in an era in which, Williams said, "I mean, the depth in the game is so much deeper now than it was (in 1997, when she started)".
But then the performance of Williams was proving pretty remarkable, too.
Despite the absence of Serena Williams, the year's concluding Grand Slam tournament could yet produce an all-American semi-final line-up. The Czech was up 2-1 in the first set courtesy a break, but surrendered the set, losing her own service twice. I didn't want to hope.More news: Harvey's impact on insurance rates in the Mid
Kvitova and Williams have played three sets in each of their five career meetings. But then Kvitova started to zone in and, punctuated by a big roar of "Pojd!" and a spirited fist pump, fired two aces from 30-all to take the game for 4-1 in the second set. The Estonian reached her sixth Grand Slam quarterfinal with a dominating victory over Daria Kasatkina on Monday.
A big test followed as she was forced to hold from 0-40 down but minutes later she double faulted to allow the American to level at 3-3. "It means a lot".
In her entirely different circumstances, Williams will walk out as a marvel.
The American had to dig deep in the ninth game, but found a 120mph ace to help her through two deuces to 5-4.
Although the match had been close throughout, the tie-break was mostly one-way traffic.
37-year-old Williams, became the oldest semi-finalist in US Open history at her age, since Martina Navratilova at Wimbledon in 1994, won the 2000 and 2001 US crowns.
Two more Americans can join Williams and Stephens in the last four with Coco Vandeweghe and Madison Keys in quarter-finals action on Wednesday.
But four Americans in the semifinals?
Venus Williams, 37, reached the finals at the Australian Open and Wimbledon after not appearing in a final since 2009. Can she really do it again? Why not. Each player has demonstrated magnificent form thus far at the Open.