Seventh seed Kvitova, twice Wimbledon champion, is playing in her second tournament after a lengthy layoff with a hand injury, suffered during a knife attack at her home in December.
But she has shown at Edgbaston that she is building up her form nicely before her return to Wimbledon, where she triumphed in 2011 and 2014.
"Both of us play really fast on this surface, the grass suits her game as well, so it was about a few points here in three, one or two key breaks in the match," Kvitova said.
It took a second-set tiebreak and one hour 43 minutes for Kvitova to secure the win which sets up a semi-final clash with compatriot Lucie Safarova.
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Safarova came through a marathon battle with Australian Daria Gavrilova, triumphing 6-7(4) 6-3 7-6(5).
Safarova saved three match-points in the deciding set and rallied to win the last four points of the third set tiebreak to end the three-hour contest on top.
"It was such an amazing fight between the both of us and I am so happy that I won," Safarova said.
"I stayed positive until the end. I had a few opportunities but she was serving well. I just believed that I could pull it out and win it," she added.
Muguruza advanced to a semi-final against Australian Ashleigh Barty, who also had a walkover, with Italian Camila Giorgi retiring with a leg injury in the first set when trailing 5-2.
Vandeweghe turned her ankle going for a shot and while she tried to continue she was clearly in discomfort.
(Reporting by Simon Evans, editing by Ed Osmond)
