Despite losing the first set in just more than half an hour, Barty hit back to seal a 4-6 7-6 (7-5) 6-2 win over the Ukrainian, who is seeded No.4 at the tournament in Germany.
The Australian will meet No.5 seed Aryna Sabalenka in the final after the Belarusian later saw off Simona Halep in straight sets.
"It's important to believe. Regardless of the score, I always try and find the right way," Barty, who also lost the opening set of her quarter-final against Karolina Pliskova, said.
"She is one of the best competitors on the tour, so I knew it would be a very hard fought match and the margins would decide it," Barty added before being presented with a bouquet of flowers with "Happy Birthday" played over the arena's sound system.
Barty had little time to celebrate the win or her birthday as she had to partner Jennifer Brady against Vivian Heisen and Wang Yafan in the doubles semi-final.
However, the Australian was seeking no sympathy despite the quick turnaround.
"I love to play tennis and I love to compete, so to do it on my birthday is even more special," Barty, who has already won the Melbourne and Miami WTA tournaments this season, said.
Sabalenka is into the second clay-court final of her career after powering past Halep for a 6-3 6-2 win after converting the first of her three match points with an ace.
"I didn't have enough sleep and I didn't want to stay on court too long," Sabalenka, who beat Anett Kontaveit just before midnight in her quarter-final, said.
"It was a good match from me and I prepared well."
Sabalenka has won three of her six games with Barty, but lost their most recent meeting in the Miami quarter-finals in early April.
"She's in form and will be at 100 per cent, I'll do my best and see how it goes," Sabalenka said.