Alston's double was the difference between the two Scottish First Division sides in a tense affair at New Douglas Park.
The game began with both teams working each other out and it was Steven Pressley's side who took the lead just before the interval when Thomas Grant's cross found Alston, who clipped the ball past the Hamilton goalkeeper and into the net.
The Bairns doubled their advantage in the 71st minute when Alston grabbed his second goal.
The ball bobbled out to the midfielder from a corner and Alston fired it into the top corner from 25 yards.
Andrew Ryan pulled a goal back for Hamilton soon after, converting the rebound from Gary Fisher's shot, but the game ended on a sour note for Accies with defender Jonathan Page receiving a red card following a poor challenge.
Falkirk will now be in the hat for the semi-final draw that takes place on Monday.