Konta produced another powerful performance to win 6-4 6-2 against the Russian qualifier, who is ranked 270th.
Konta will face Serena Williams or Greek 13th seed Maria Sakkari in the last eight.
That will represent a step up in a class of opponent for the Briton, although Konta should not fear either player on the evidence of her two matches in this event.
Again she hit confidently and cleanly against Zvonareva, a two-time Grand Slam finalist who is trying to reach back up the rankings after taking time off tour to give birth in 2016.
There were plenty of positive signs for Konta as she builds up to next week’s US Open in New York.
“She is very tricky to play, she has been around a long time. I served well and I was building the points well where I could,” said Konta.
The Western and Southern Open is held in Cincinnati but was moved to Flushing Meadows to create a two-tournament safety bubble that also incorporates the Grand Slam event, which starts on Monday.
Former world number one Murray is the only other British player left in the singles draws. He will be bidding for a place in the men’s quarter-finals on Tuesday when he takes on Canada’s Milos Raonic.