The world No. 8 and top seed made a superb start in Nottingham but was pegged back and then edged out in a tight finish to lose 2-6, 7-6, (3), 7-5 to the Croatian.
Konta looked to have overcome a mid-match wobble as she recovered to lead 5-4 in the final set but Vekic struck back to claim her second singles title.
"All credit to Donna, she was continuously able to raise her level all throughout that match," Konta said on 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 broke Vekic in the first game of the match and powered through the opening set in commanding fashion.
Vekic's recovery started as she broke to lead 3-1 in the second set. Konta immediately broke back but there were signs of frustration as she disputed a line call.
She lost the set on a tie-break and there was further anguish early in the final set as she contested another decision.
Again she fell 3-1 behind but after recovering to draw back level and then lead, it seemed she held the edge.
But after serving to stay in the match, Vekic claimed a crucial break and then served out for victory.
Vekic said: "It's pretty amazing. It's my first grass-court title and my second overall. I'm really happy.
"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."