American Gauff will face the winner of the second semi-final, to be played later in the day between Paris Olympics gold medallist Qinwen Zheng and Karolina Muchova, in Sunday's final.
"It was just a tough match... I just tried to stay focused. Mentally I tried to reset. I got down a couple of times but I tried to bounce back," Gauff said in her post-match interview.
Double faults and a weaker second serve troubled world number six Gauff early in the match as Badosa won 75% of her second return points in the set.
Badosa broke to go 3-2 up but Gauff fought back, making it 4-4 after a marathon game where Badosa saved 10 break points but a persistent Gauff ultimately prevailed.
The world number 19 re-took the lead with a quick break in the very next game and held her own serve to win the set.
Badosa broke again in the first game of the second set and almost built a 4-1 lead but fourth seed Gauff saved four break points to hold.
A confident Gauff would not lose another game in the set as she seized the momentum, landing more precise shots across the court, with Badosa not winning a single point as Gauff broke her in the last game.
In the third set, the American broke twice to build a 4-1 lead, hitting two aces as her powerful forehands eluded an increasingly exhausted Badosa.
Gauff sealed victory with another break.
"I had tough opponents the whole tournament. Happy to be in the final," she said.
