The Bairns appeared to be suffering a Scottish Cup hangover after surrendering a three-goal lead to Hibernian in Saturday's semi-final at Hampden to lose 4-3 after extra-time.
Mark Gilhaney put Dumbarton ahead on 19 minutes when he cut inside a defender and dispatched the ball into the bottom corner.
Falkirk were still reeling from that blow when Garry Fleming (21) finished a superb passing move to double the lead.
Lyle Taylor pulled a goal back for the Bairns 12 minutes later with a glancing header to lift the spirits of the home supporters.
Fleming added his second to settle the nerves when he kept his composure the slip the ball past the advancing keeper on 65 minutes.