Birrell, the world number 60, got off to a slow start in Seoul, losing her opening service game to find herself 0-2 behind.
But she soon righted the ship, taking the next four games before tidily serving out the first set.
Joint, ranked 99th, collapsed in a scrappy second set, losing all four games on serve to open a path for Birrell's triumph.
The top seed, former French Open champion Jelena Ostapenko, fell to another Australian, Taylah Preston, in the round of 16.
Birrell was seeded second, and Joint was seeded fifth.