Storm Hunter and Adam Walton also made early exits on day one at Flushing Meadows as Gibson was unable to disguise her anguish at an opportunity lost.
The 22-year-old rallied from a set and a service break down to look poised to claim her maiden grand slam victory outside Australia before Vekic steadied to post a 6-2 4-6 6-3 in two hours and 33 minutes.
The veteran Croat and 2024 Wimbledon semi-finalist seemed to be unravelling after blowing a set and 4-1 lead, cursing herself and berating her box as Gibson roared back into the contest.
But Gibson was unable to wrestle back control after failing to convert any of four break-point chances in the second game of the deciding set.
She had Vekic down 0-40 serving at 4-2, only to miss five straight returns to ruefully let the Paris Olympics silver medallist off the hook.
"It's a hard one to swallow," Gibson said.
"Most of the time it's who makes the most unforced errors on the court who comes away with the loss and today that was me, so it's just something to learn from."
But after starting the year well outside the world's top 100, Gibson has slashed her ranking in half after racking up some 30 match wins, including over former No.1 and dual Australian and US Open champion Naomi Osaka.
"Days like these are really tough. It's hard not to be disappointed. When it comes to the slams, every player wants to do well," said the world No.63.
"But if I'm looking at my entire year so far, I've been able to achieve some things that probably at the start of the year I probably wouldn't have thought I'd be able to do.
"So to still be in the position ranking that I currently have and moving forward to these next tournaments, I don't really have any points to defend, so it's kind of just no pressure and more opportunities."
Walton missed two set points in the third set of a tight 7-6 (7-2) 6-2 7-5 loss to China's Yibing Wu before Hunter, after a taxing run through three qualifying matches, ran out of gas in a 6-4 6-1 defeat to 11th-seeded Ukrainian Marta Kostyuk.Â
Kim Birrell faces Petra Marcinko later on day one, with Taylah Preston taking on American Alycia Parks.
Aleksandar Vukic, up against Japan's Rei Sakamoto, is the only other Australian in men's action.
World No.7 and three-time US Open quarter-finalist Alex de Minaur opens his title quest on Tuesday against Italian qualifier Andrea Guerrieri.
This AAP article was made possible by support from Tennis Australia.
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