Oliva hit a Games record 45 of her 50 targets to win Guatemala's first Olympic gold in any sport ahead of Stanco, who hit 40.
Australia's Penny Smith could not make the gold medal round but hit 32 of her 40 targets to finish third.
Oliva began as a promising gymnast, who represented her country at the 2010 Pan American Championships but had to quit the sport after the injury.
It took her one year to recover and her doctor suggested she tried shooting instead.
She began trap shooting in 2013 and worked as a volunteer at the 2016 Rio Games before eventually making her Olympic debut in Tokyo, where she came 26th.
"When I was younger, I was dreaming about (competing in) Olympics in gymnastics," she said.
"I never thought I'd excel in this sport. It was all new to me.
"It was only when I started viewing and visiting shooting events that I thought it would be possible to represent Guatemala in Olympics Games.
"Lately, my dream was to win the gold medal in Olympics."
It has been an incredible 24 hours for Guatemala, which had just one Olympic medal heading into Paris.
Jean Pierre Brol won bronze in the men's trap on Tuesday before Olivia struck gold on a hot afternoon at the Chateauroux Shooting Centre.
The Rafael Nadal fan got off to a strong start and was flawless until her 17th shot when she missed her first target.
She hit 23 out of 25 to lead the six-strong group heading into the elimination round of an event in which shooters take aim at flying clay targets from their stations.
Stanco and Smith overtook Spaniard Mar Molne Magrina but Oliva never looked under pressure.
"I can't even believe it’s for real, I’m so happy," she said.
"I hope more people can do the same or more for the sport.
"We only have one (shooting) range, so I hope this will make the sport bigger and more people will try it."
Tokyo champion Rehak Stefecekova of Slovakia could not make the final, while world number one Fatima Galvez of Spain came fifth.