Konta appeared close to winning by default at the end of the first set when Vekic received treatment for a back injury before the rain forced both players off court to give Vekic a slight respite.
But Konta was not to be denied and overcame a couple of minor wobbles to win through. "It is never easy with the rain delay and it was definitely challenging," said Konta.
Barty, who lost to Konta in this event last year, took just over an hour to complete her routine win on Saturday, breaking Osaka's serve three times and winning 80 percent of points on her own service.
"I think it's been a very clean week for me. I've served well and gotten myself into return games very well," Barty said.
It will be the 22-year-old Australian's first final since losing out to Angelique Kerber in the Sydney International in January.
(Reporting by Neil Robinson, editing by Pritha Sarkar)
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