The Croatian, ranked 37th in the world, came back from a set down under the roof on Court No. 1 to win 5-7, 6-4, 6-1 and reach the last four at a Grand Slam for the first time.
She will meet Italian seventh seed Jasmine Paolini or US 19th seed Emma Navarro for a place in Saturday's final.
Sun, ranked 123rd in the world, drew first blood, breaking in the 11th game and serving out for the first set.
Vekic battled her way back into the contest, but her serve fell apart as she served for the second set at 5-3, producing five double faults.
The Croatian immediately regrouped and broke Vekic in the next game to take the set 6-4.
From then on it was one-way traffic as the 28-year-old made it 16 points in a row, racing into a 5-0 lead and wrapping up the match in a little over two hours.
Sun, who came through the qualifying competition to reach the Wimbledon main draw, was aiming to become just the second woman from New Zealand to reach a Grand Slam semi-final, following Belinda Cordwell, who made the 1989 Australian Open semi-finals.
