The 27-year-old Bacsinszky won 6-4, 7-5 in her first ever meeting with Aryna Sabalenka in one hour 42 minutes.
The 27-year-old Bacsinszky won 6-4, 7-5 in her first ever meeting with the Belarus teenager in one hour 42 minutes.
"I wasn't sure I could win today until the last ball in play," Bacsinszky said.
"Aryna is a very good player with excellent psychological stability, which is a rare feature at her age."
Bacsinszky, who is currently 22nd in the world, broke twice in the opening set for a one-set lead in 44 minutes, while Sabalenka, ranked 125, broke only once.
Sabalenka led 3-1 and 5-3 in the second but Bacsinszky levelled on both occasions before producing the deciding break in the 11th game minutes before she took the victory.
Earlier on Saturday, Aliaksandra Sasnovich, ranked 96th, put Belarus into the lead beating Viktorija Golubic 6-3, 5-7, 7-5.
"It was a very hard match and I'm so happy to bring my team the first point," Sasnovich said.
Golubic, 54th in the world, started with an immediate break but the 23-year-old Belarusian broke back twice to take the opening set in 40 minutes.
In the second, Golubic took a 4-0 lead before Sasnovich rallied to level at 5-5. The Swiss player broke again to level at one set all.
In the decider, Sasnovich produced the deciding break in the 12th game to chalk up Belarus' first point.