Rybakina, the 2022 Wimbledon champion, went down an early break in the Group D tie but cranked up her powerful forehand and drew level after six games before claiming the opening set on serve with a big ace at the end.
Birrell raced to a 5-1 lead in the next but squandered a set point, and Rybakina clawed her way back with some ferocious shot-making to clinch victory via the tiebreak.
"It was such a difficult match," said Rybakina, who secured her eighth victory in 10 singles matches since her debut in the competition in 2021.
"I'm super happy to bring the win for the team... it's always nice to be back in Australia."
Earlier, Yulia Putintseva defeated debutant Maya Joint 6-2, 6-1 at Pat Rafter Arena to give Kazakhstan the early lead in the best-of-three matches contest.
However, Australia's Storm Hunter and Ellen Perez claimed the third match of the night with a 6-3, 6-4 victory over Anna Danilina and Zhibek Kulambayeva in the doubles.
Other BJK Cup results
Czech Republic beat Brazil 2-1 in Ostrava in Group B, while the Netherlands cruised past Germany 3-0 in The Hague in Group F.
Poland, without French Open champion Iga Swiatek after the world number two opted to skip their Group E ties to focus on her training ahead of the clay season, beat Switzerland 3-0 in Radom.
Jessica Pegula and Danielle Collins were among a trio of players to withdraw from the United States team, which plays Denmark on Saturday in Group C, for the three-day qualifiers.
Teams have been divided into six round-robin groups of three in the qualifiers this year, with the winners joining hosts China and 2024 champions Italy at the Finals in Shenzhen towards the end of the season.