Konta took the first set, but the world number eight couldn't seal the silverware in the second set tie-break.
Konta took the first set, but the world number eight couldn't seal the silverware in the second set tie-break and eventually slumped to a 2-6, 7-6 (7/3), 7-5 loss.
The 26-year-old's failure to land her first grass-court title is a setback ahead of Wimbledon, which gets underway next month.
Sydney-born Konta has never been past the second round at Wimbledon, with her best Grand Slam performance a run to the semi-finals of the 2016 Australian Open.
"All credit to Donna, she was continuously able to raise her level all throughout that match," Konta told Eurosport.
"I was not able to stay with her. Full credit to her for playing incredibly well."
"But I'm very happy that I got to play five great matches on the grass. For Donna and I, heading into Wimbledon, we just want to play on the surface as much as possible."
Thanks to her gutsy effort, 20-year-old Vekic, ranked 70th in the world, was able to celebrate her second WTA singles title.
"It's pretty amazing. It's my first grass-court title and my second overall. I'm really happy," she said.
"Jo played amazing, she's having an amazing year, but I was just trying to focus on myself and serve as good as I can and be aggressive."