World number 40 Vekic denied her opponent the chance to earn a first grass title on home soil, winning 2-6 7-6(3) 7-5 on Sunday afternoon in the searing heat.
It was a big disappointment for Konta, who recorded a straight-sets victory over Magdalena Rybarikova in the semis to keep her hopes alive in the East Midlands, but she is happy with the momentum built up heading into Wimbledon.
"All credit to Donna, she was continuously able to raise her level all throughout that match," she 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."
Konta has never previously made it past the second round of Wimbledon, which gets up and running on July 3.