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 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.
"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.
"I had tough opponents the whole tournament. Happy to be in the final."
In the second semi-final, Muchova was fully in control against Paris 2024 Olympics gold medallist Zheng Qinwen, who suffered a hard fall in the opening set and took a lengthy medical timeout.
Zheng provided a degree of resistance in the second set, but 49th-ranked Muchova took a 4-2 lead before sealing her place in a sixth WTA final with a 6-3, 6-4 victory over the Chinese fifth seed.
Muchova, in only her seventh tournament back after suffering a wrist injury at the 2023 US Open, continues to put on impressive performances.
The Czech reached the US Open semi-finals in September and on Friday ended top seed and three-time Grand Slam champion Aryna Sabalenka's 15-match winning streak in the China Open quarter-final.