Bacsinszky took full advantage of 50 unforced errors from 19-year-old Sabalenka to win 6-4, 7-5.
Sasnovich's aggressive style, with 36 winners to Golubic's 22, helped her edge an almost three-hour fight which ended 6-3, 5-7, 7-5.
With former top-ranked Victoria Azarenka taking a break to raise her first child, Sasnovich is the only member of the Belarus team ranked in the top 100, at 96. Golubic is 54th.
Sasnovich plays Bacsinszky in Sunday's first reverse singles, then Sabalenka takes on Golubic. The tie will conclude with a doubles rubber between Belarusians Olga Govortsova and Vera Lapko and the Swiss duo of Martina Hingis and Belinda Bencic.
The Swiss are aiming to reach their second Fed Cup final following defeat to Spain in 1998, while Belarus is in uncharted territory in its first semifinal.