Radwanska, seeded nine, looked strong in the first set but lost seven consecutive games to allow her opponent - ranked 37 in the world - to take a lead in the third.
Pliskova and Barty have split two prior meetings-both on Nottingham's grass-with three of the five sets they've played decided in tie breaks.
Seventh-seeded Dominika Cibulkova of Slovakia, ninth-seeded Agnieszka Radwanska of Poland and 16th-seeded Elena Vesnina of Russian Federation breezed to straight-set wins while eighth-seeded Jelena Ostapenko of Lativa needed three sets to dispatch Czech Barbora Strycova, 2-6, 7-5, 3-6.
The 2015 Wuhan runner-up, Muguruza will maintain her hold the top spot if she beats Ostapenko and reaches her eighth semifinal of the season.
Muguruza, who required treatment to that leg during a straight-sets win over Lesia Tsurenko on Tuesday, was very much second best in the second set as Polish outsider Linette hit back.
"It was a very tough match and Monica was playing very well, but I'm really happy to win the match", Ostapenko said after the match.
Barty, who had already downed fifth seed Johanna Konta in the second round, is now in the running to qualify for the WTA Elite Trophy in Zhuhai.
Muguruza advances to face Poland's Magda Linette, who coasted to a 6-1, 6-2 win over Katerina Siniakova of the Czech Republic. Ostapenko won an up-and-down three set match, but she can clean her game up quite a bit moving forward.
Greek qualifier Maria Sakkari surprised No. 16-seeded Elena Vesnina, 7-6 (6), 7-5, one round after she toppled fourth-seeded Caroline Wozniacki. The top seed broke early for a 2-1 lead, and with Ostapenko spraying errors all around the court, the Spaniard was able to secure a double break lead and holding for a commanding 5-1 advantage.
The Slovakian was beaten 6-3 7-5 by Caroline Garcia, continuing the Frenchwoman's impressive season.