Fritz was well on track to win Saturday night's clash in Perth after taking the first set and going up 2-0 in the second.
But world No.45 Baez had other ideas, unleashing his trademark fighting spirit to turn the match on its head.
Baez took the second set in 66 minutes and snared an early break in the third on the way to his 4-6 7-5 6-4 victory in two hours and 33 minutes.
The win gave Argentina a 1-0 lead, meaning world No.3 Coco Gauff needs to beat Solana Sierra in the women's singles to keep the US alive in the tie.
Argentina beat Spain 3-0 on Friday, meaning a win over the US would guarantee their place in the quarter-finals.
Fritz's frustrations boiled over after losing the final point, the American snapping his racquet over his thigh.
After shaking hands with his opponent, a downcast Fritz then gave the broken racquet to a kid in the crowd before leaving the court.
Baez had lost all five of his previous meetings with Fritz, and he was over the moon to finally take down his highly-fancied opponent.
"Wow, I'm so happy right now," Baez said.
"We are searching this feeling for many years. Taylor beat me many, many times, so I'm happy to take one."
On the eve of the United Cup, Fritz revealed he was still dealing with "pretty serious tendonitis" that had plagued him at the tail end of last year.
The American knows he will have to deal with the knee issue at this year's Australian Open, and his loss to Baez is sure to raise some early alarm bells.
Fritz wrapped up the first set in 38 minutes, and he broke his opponent early in the second set before Baez fought back with a vengeance.
Baez's energy grew as the match wore on, while the opposite was true for Fritz, who became increasingly frustrated at his own unforced errors.
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