The 24-year-old, who struggled with a back injury last season, claimed the first set 6-3 before Cornet prevailed by the same score in the second to force the decider in the all-French encounter.
Garcia appeared to be suffering from some discomfort and took a medical timeout before the final set. She tried to hit a couple of serves before deciding she could not continue and retired in tears.
"I've always had some issues with my back," Garcia told reporters.
"At the beginning of the match I felt a pain in my lower back and it was different than it is normally. It was getting worse and worse and in the end I couldn't really move. There was no point to keep going."
She was hopeful the injury would not affect her Australian Open prospects next month.
"I mean, it's in two weeks, so it's a long time... with the back it's always complicated," she added. "Especially mine, it's really something I take care of, so it's kind of surprising for it to show up like this. But we'll do the best that we can."
Cornet will met Mirjana Lucic-Baroni in the second round after the Croatian, who travels full-time with her daughter, overcame Germany's Tatjana Maria 6-4 4-6 6-0.
"I'm very sorry for Caroline. I know what it's like to have back pain," Cornet, who went down to Karolina Pliskova in last year's final, said in a courtside interview.
"I hope she recovers for the next tournament in the Australian Open. I cannot say I'm glad to be through this way, but the adventure continues for me.
"Last year, I had a super run to the final and I have very good memories. I hope to go as far as possible."
Elsewhere, seventh seed Anastasija Sevastova beat local favourite Sam Stosur 6-1 6-3, while Ana Konjuh tamed Kiki Bertens 6-1 6-2.
In the men's section, Ukraine's Alexandr Dolgopolov caused a minor upset with a hard fought victory over sixth seed Diego Schwartzman of Argentina 6-4 7-6(3).
(Reporting by Amlan Chakraborty in New Delhi; Editing by John O'Brien)
