Konta, 27, reached the last eight by beating compatriot Heather Watson on Wednesday and she made a relatively slow start to Thursday's match when Jakupovic broke in the third game.
However Konta regrouped and she found back-to-back breaks at the end of the set to take it 6-4.
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And although Jakupovic broke her in the first game of the second set Konta broke back immediately and took control of the match to easily reach the semi-finals.
There she will face Croatian Donna Vekic, who beat her in the final 12 months ago.
Konta said on Eurosport: “It was very tricky out here, quite windy, and I’d never played her before. The most important thing is that I’m competing well and fighting for every single point. I’m definitely enjoying getting more matches under my belt and we’ll work towards getting another one.”
Following their clash here 12 months ago, Konta and Vekic met in the second round of Wimbledon, with the British player coming through 10-8 in the deciding set.
“We’ve had some incredible battles, especially the last two times,” said Konta. “She’s playing very well on this surface. I’m looking forward to the challenge, it’ll be a great test for me to keep competing the way I have and keep raising my level.”

Johanna Konta of Great Britain celebrates breaking Heather Watson of Great Britain in the second set during Day Six of the Nature Valley Open at Nottingham Tennis Centre on June 14, 2018 in Nottingham, United Kingdom.

Image credit: Getty Images

British number four Katie Boulter was making her first appearance in a WTA Tour quarter-final but found top seed Ashleigh Barty too savvy, the Australian winning 6-0 6-2.
In the men’s event, which is on the second-tier Challenger Tour, Dan Evans reached his second successive semi-final with a 7-6 (7/5) 6-3 victory over Canadian qualifier Brayden Schnur.
The former British number two is making the most of his opportunities on his favoured grass as he works his way back from a drugs ban.
In a feisty contest, Evans saved a set point in the opening set before claiming the only break of the match to set up a clash with Spain’s Marcel Granollers.
The 28-year-old, who is waiting to find out whether he will be given wild cards for the big grass-court events, said: “I’ve been working pretty hard off the court as well. I’ve done everything I can to be in this position now.
“You can come here and easily lose in the first round, there are some great players here. It’s been two good weeks.”
Evans was one of two British players hoping to make the last four but James Ward fell 6-4 6-3 to Ilya Ivashka.
Additional reporting via PA Sport
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