The 22-year-old Florida-based Puig defied her ranking of 34 to get the better of the world number two and Australian Open champion on the back of a sizzling 54 winners yesterday.
Puig, who also knocked out French Open champion Garbine Muguruza in the quarter-finals, is the first woman from the Caribbean island to win a medal of any sort.
Before her golden breakthrough, Puerto Rico had won just eight Olympic medals, two silver and six bronze. Six of that total had come in boxing.
"Oh my God," she screamed before breaking down in floods of tears and parading her national flag around centre court.
"It's amazing. My country appreciates this and I give this victory to them. The way I did it, I wouldn't want it any other way," said Puig who has lived in Florida for most of her life.
"There was a lot of relief and tension. The ball which went out seemed to be moving really, really slowly. I wished it had sped up a bit more."
"I can't believe what I have achieved. It's one of my biggest dreams."
Kerber had cruised into the final without dropping a set as the second seed aimed to emulate Steffi Graf who was Germany's last champion at the 1988.
However, her perfect record in Rio ended when Puig grabbed the first set against a suddenly lacklustre looking German who required treatment on a back injury between sets.
Kerber, 28, levelled the final in the 10th game but needed four set points to get the job done.
That set the foundation for the thrilling decider where Puig raced into a 5-0 lead before having to suffer for her place in history during a marathon seventh game.
"She deserved to win," said Kerber, a Wimbledon finalist last month.
"It's not the medal I wanted but it's still a silver. She was hitting everything, she played a great match.
On Friday, after beating two-time Wimbledon champion Petra Kvitova in the semi-finals, Puig had vowed to win gold and bring some light relief to Puerto Rico.
The US territory is buried under some $70 billion in debt.
There is also a growing concern over the Zika virus, which has spread quickly through the island.
In July, US health officials reported that as many as 50 pregnant women per day have been infected, raising the risk of birth defects.
Before London in 2012, all of the island's Olympic medals had come in boxing.
Meanwhile, Kvitova won the bronze medal play-off by defeating American seventh seed Madison Keys 7-5, 2-6, 6-2.
In an untidy clash, the two women hit a combined 87 unforced errors while Keys's challenge was undone by only converting two of her 10 break points.
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