Andreeva, the 19-year-old fourth seed, is the favourite for the tournament in Beijing now that a series of upsets have knocked out her more esteemed rivals.
The French Open champion dropped just three points to grab a two-game lead but still needed a tie-break to edge 28th-ranked Alexandrova in the first set.
Both players struggled to extend rallies in a sparsely populated stadium as they tumbled into another tie-break.
Andreeva found the breakthrough with a masterful forehand to go 3-2 ahead, then held her nerve, dropping just one more point before crossing the finish line.
"I am very happy that we finished this match, because I knew if it (went) to the third set it's going to be really tough for both of us," Andreeva said.
"She played unbelievable," the teenager said of her 31-year-old opponent. "What a match."