The tie was delicately poised at 1-1 heading into the second leg at New Douglas Park following Tuesday's stalemate.
However, Anthony Andreu's first-half goal proved enough for the Accies to win the tie as Alex Neil's side now face another two-legged affair with Hibernian, who finished second from bottom in the SPL.
The hosts took a big step toward ending their three-year absence from Scotland's top flight after 15 minutes. Jason Scotland skipped past David McCracken before squaring to the unmarked Andreu, who tapped home from close range.
Falkirk continued to threaten after the break and were unfortunate not to draw level in the tie as top goalscorer Rory Loy saw his would-be equaliser cleared off the line early in the half.
The miss was academic, though, as Hamilton held out to book a tie with Hibernian and move nearer to securing a first top-flight return since 2011.