The world number 19's recent run of early tournament exits looked in danger of continuing when Australia's Gibson, ranked 38, took the first set at the Pan Pacific Open.
But Eala dug in and evened the match before hitting top gear in a dominating third set.
The 21-year-old said the turnaround had been "a gradual increase of level."
"Maybe just a couple of small details that, when accumulated, make a big difference," she said.
"In one 30-all I went more for my shot or in one deuce I did so and so.
"Those little details, when all summed up, that's the difference."
Eala faces Russian third seed Diana Shnaider in the quarter-finals.
She became the first player from the Philippines to win a title on the women's WTA Tour in August, and her every move is big news at home.
Eala was again backed by passionate support in Tokyo, but she was on the ropes in the first set as Gibson punished her for missing chances.
She began her comeback in the second set, but she still let her opponent off the hook on several occasions.
Eala went from strength to strength as the match moved into the final set and closed it out when Gibson dumped a shot into the net.
She hit nine aces and broke Gibson's serve six times in the match.
"Of course the support helps a lot all the time," said Eala.
"It creates a great atmosphere, and I think there was a bit of back and forth within the crowd.
"You want to be present; you're very invested in the match, so it comes naturally to focus."
Shnaider beat Aliaksandra Sasnovich of Belarus 6-3, 6-7(5), 6-3 to book her place in the quarter-finals.
World number one Elena Rybakina plays her opening match on Thursday when she takes on fellow Kazakh Yulia Putintseva.
Home favourite Naomi Osaka is also in action against Slovakian wild card Mia Pohankova.