Vekic is no slouch on grass and Barty, emerging as one of the favourites for Wimbledon, said it was the "perfect test".
"I knew that I had to be sharp and I had to be ready. Otherwise it wouldn't even be a contest," world number two Barty said on court.
"It was nice and clean today out here. I was just really excited to come back out and kind of dust off the cobwebs a little bit."
Unseeded Jelena Ostapenko, who won the French Open two years ago but has struggled this year, ended the hopes of Britain's Johanna Konta, winning 6-3 6-4.
Five-time Wimbledon champion Venus Williams also moved through with the 39-year-old defeating Aliaksandra Sasnovich.
Barty, a former junior Wimbledon champion, dropped serve early on but her deep, skidding sliced backhand began to work its magic as the Vekic errors mounted.
She broke twice to take the opening set and twice more to seize control in the second.
Vekic prolonged the contest to grab back one of the breaks when Barty served for the match. However, Barty made no mistake the second time around when she brought up match point with an ace and sealed victory when Vekic netted an attempted drop shot.
Barty's French Open title was somewhat unexpected seeing as she is not a renowned claycourter. Grass, on the other hand, is tailor-made for her game which is why all eyes will be on her at Wimbledon which starts a week on Monday.
The odds are against her following up her French title by winning Wimbledon. Only six women -- Margaret Court, Billie Jean King, Chris Evert, Martina Navratilova, Steffi Graf and Serena Williams -- have managed that in the professional era.
She will continue to hone her grasscourt skills against American Jennifer Brady who beat Lesia Tsurenko 6-3 6-3.
(Reporting by Martyn Herman; Editing by Christian Radnedge)
