The unseeded Williams fought back from 5-3 in the final set before serving out the victory.
Konta had won 21 of her last 22 matches going into Thursday's match, including a victory over top-seeded Simona Halep on Wednesday.
''Johanna played so well. She's had a wonderful summer and beat so many players this tournament,'' Williams said. ''It seemed like I was finished but I felt the energy of the crowd behind me.''
The former top-ranked American will next face U.S. Open finalist Roberta Vinci, who beat Karolina Pliskova 7-6 (2), 6-3. The 15th-seeded Italian saved three of four break points and took advantage of 30 unforced errors by the eighth-seeded Pliskova.
Williams, who earned her 700th career victory to reach the quarterfinals, has never dropped a set against the Italian veteran - the same woman that ended little sister Serena's chances of a calendar-year Grand Slam in the U.S. Open semifinals.
''You don't play at this level every day, so for me it was great to be playing like this,'' said Muguruza, who converted five of her seven break opportunities.
Muguruza will next play Angelique Kerber, who advanced when Coco Vandeweghe retired while trailing 6-3, 3-1.