Russian third seed Vesnina needed one hour and 36 minutes to dispatch Puerto Rican Puig 6-4 6-4 and set up a meeting with qualifier Estrella Cabeza Candela.
There was disappointment for Peng Shuai as the former world No 14 crashed out in straight-sets to home hopeful Storm Sanders.
The 19-year-old Australian No 7 will now face second seed Kirsten Flipkens after shocking Peng 6-2 6-2.
American Alison Riske became the first woman to reach the quarter-finals after recovering from a mid-match wobble to oust Perth-born Casey Dellacqua 7-5 4-6 6-3.
Riske will be joined in the last eight by Bojana Jovanovski after the eighth seed from Serbia battled back to beat Olivia Rogowska 3-6 6-2 6-0.