World number 17 Barty has won just one WTA Tour title, but she is well placed for a challenge this week after another straight-sets triumph.
The Australian defeated Ying-Ying Duan 6-3 6-2 to set up a last-eight meeting with British wildcard Katie Boulter, who saw off Sam Stosur.
And Boulter is not the only homegrown talent still in the draw, with Konta battling past compatriot Watson 6-4 7-6 (7-5). Defending champion Donna Vekic also progressed by seeing off Vera Lapko 6-2 7-6 (8-6).
The Rosmalen Grass Court Championships saw only one WTA match completed, with rain halting play before Elise Mertens and Antonia Lottner could complete the first set of their clash.
Alison Riske was the sole player to advance by beating Fanny Stollar 7-6 (7-1) 6-2, setting up a quarter-final meeting with top seed Coco Vandeweghe.