The 18-year-old Russian, who won back-to-back WTA 1000 titles last year in Dubai and Indian Wells, will meet Paris Olympics silver medallist Donna Vekic in the opening round at Melbourne Park.
Mboko, 19, raced to a 3-0 lead but Andreeva turned the tables to reel off the next six games to take the opening set.
It was one-way traffic in the second set as Andreeva broke early to take the opening three games and sealed the match on serve, setting herself up perfectly for the year's opening Grand Slam, which begins on Sunday.
Canadian Mboko took a medical timeout midway through the match. The world number 17, who became the youngest player since Serena Williams to beat four major champions in an event on the way to winning last year's Canadian Open, faces Australian Emerson Jones in her Melbourne Park opener.