"That was a tough match - I knew she was going to play well because she's already had a few matches on grass," Vekic, the sixth seed, said.
"It was my first match so I was struggling a little bit to find some rhythm, but I think winning 7-6 in the third is very good for my confidence."
Barty, a quarter-finalist at the tournament in 2016 and 2017, proved why grass was her favourite surface as she swept aside Switzerland's Stefanie Voegele 6-3 6-3 to progress to the last-16.
"It was pretty clean for a first match on grass. I'm very pleased with that," Barty said. "The beauty of grasscourt tennis is that you get better with matches, and there's no better way than to get out there and try to fix up your game."
Fourth-seed Konta, who finished runner-up to Vekic at Nottingham last year, is looking to move past an underwhelming start to the season in front of the home crowd. The 27-year-old beat Japan's Kurumi Nara 6-2 6-3 to book a second round spot.
Konta will take on compatriot Heather Watson in the last-16 while Barty faces China's Yingying Duan.
Slovakia's Magdalena Rybarikova, Japanese third seed Naomi Osaka and Romanian Mihaela Buzarnescu will play their second round matches on Wednesday.
(Reporting by Aditi Prakash in Bengaluru; Editing by Toby Davis)
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