The wicketkeeper, who ended up scoring 106 and celebrating his first Ashes hundred at his home Adelaide Oval, later admitted to feeling a "bit of a feather" off his bat when he wafted at a Josh Tongue delivery when on 72.
Carey stood his ground when the appeal was turned down, but England were convinced and went to the decision review system.
The 'Snicko' technology picked up a sound, but it did not align with the ball's flight by the bat in the review of the delivery, so the decision stood.
Carey went on to score another 34 runs to help Australia reach 326 for eight at stumps after the hosts won the toss and elected to bat first.
England bowling coach David Saker said England were ill-served by the technology - and not for the first time in the series.
"The boys are pretty confident he hit it. I think the calibration of the 'Snicko' is out," the Australian told reporters after England toiled through a sweltering day in the field.
"It was a pretty important decision. Those things hurt. You'd think in this day and age the technology's good enough to pick things up like that."
Saker said England had been unhappy with the technology throughout the series, without elaborating on specific instances, and might raise the matter with match officials.
"After today, that may go a bit further," he said of complaints within the England camp.
"It should be better than that. It is what it is."
During Carey's press conference, he admitted that he felt he had nicked the ball, saying that he had got lucky with the decision.
After those comments, the company that runs the technology, BBG Sports, took full responsibility for the incident, indicating in a statement that it was 'human error' that led to the wrong decision.