Eala, ranked 28th, overpowered Pegula over the last two sets as play resumed in the final suspended after lengthy weather delays on Sunday.
Playing in her second final, after a runner-up finish at Eastbourne last year, the 21-year-old left-hander produced a rock-solid service display and an impressive array of winners to deny third-ranked Pegula a 12th career title and a third this year.
"I feel so much love. My first chance at a title knowing it won't be the last and already having achieved this milestone for my career," Eala said on court. "I knew whatever happened after this match would have already been a win for me."
The scoreboard showed a match time of one hour and 45 minutes, but it took nearly 24 hours to complete.
Pegula had won the first set 6-4 and Eala was up 2-1 in the second on Sunday when officials postponed the finish until Monday after lengthy rain delays.
The pair had waited through a three-hour delay before starting, and once the match was halted it was another five hours before officials called a halt for the day.
That also saw the ATP final between 10th-ranked American Taylor Fritz and 19-year-old Spaniard Rafael Jodar postponed.
Pegula opened the action on Monday with a nervy service hold for 2-2.
But she could make no inroads on Eala's serve and Eala broke Pegula in the 10th game to take the set and force a third.
The final set was all Eala, to the delight of a vocal contingent of supporters.
"I want to congratulate Alex on an amazing tournament," Pegula said.
"To see how far you've come over the last couple of years when we first played, to see the amazing fans that follow you every place you go, it's not fun to play against but I think it's amazing."
Eala had been enthusiastically cheered all week as she made her way to the final with wins over second-seeded Elina Svitolina, third-seeded former world number one Naomi Osaka and defending champion Leylah Fernandez.
Eala is now projected to rise to a career-high 20th in the world.