The 20-year-old Vekic came through a close encounter with former Roland Garros finalist Lucie Safarova 7-6 (7/5), 3-6, 7-6 (7/4), after two hours and 50 minutes.
The British number one beat Ashleigh Barty 6-3 7-5 on Friday to edge closer to her first title on home soil and third of an impressive 2017.
Playing in the semi-finals of the competition for the first time, Konta twice broke Rybarikova in the opening set.
"I'm looking to constantly improve and evolve as a player, regardless of surface", Konta said in her post-match press conference.
After swapping breaks in the second, the Slovakian double-faulted in the 11th game to gift Konta the advantage and the Briton served out the match.
The Briton is in action again on Saturday afternoon, as she and doubles partner Yanina Wickmayer of Belgium play Australian pair Monique Adamczak and Storm Sanders in the semi-finals. The decider also went the distance, Konjuh surviving five match points in the breaker before converting the first of her own to finish off an epic 6-3 6-7 (4-7) 7-6 (12-10) triumph. "I worked very hard to get myself back into that second set and stop Ashleigh from running away with it".
Konta has won three tournaments in the past year and will face Slovakia's Magdalena Rybarikova in the last four.
Having reached the top six in the world rankings last month, she is also optimistic about an improved performance at Wimbledon, where she won a round for the first time previous year after four opening-round defeats.
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