Blair Alston struck just before half-time to give the Bairns the lead against the run of play and he doubled their lead with a stunning effort from the edge of the box after 71 minutes.
Substitute Andy Ryan pulled one back for the hosts, yet they were unable to grab a leveller in the closing stages.
A disappointing afternoon for Billy Reid's side was completed when Jonathan Page was sent-off in injury-time.
Kevin Cuthbert was forced into an early stop when he tipped over Stewart Murdoch's shot from the edge of the box.
Hamilton threatened at the other end as Gary Fisher combined with Jon Routledge, who sent a shot just wide of the target.
Fisher should have taken advantage after Jonathan Flynn failed to clear a routine high ball, but he was denied by an excellent stop by Michael McGovern.
Stevie May dragged an effort wide of the target as Accies continued to threaten, though they almost paid the price for being lacklustre in possession when Alston picked up a loose pass before driving forward and firing a shot straight at McGovern.
Cuthbert was called into action twice within the space of a minute as he denied David Weatherston before comfortably gathering Alston's shot from the edge of the box.
Darian MacKinnon picked up the first booking of the match for a rough challenge on Stephen Kingsley.
Steven Pressley was forced into a first-half substitution when Kingsley limped off to be replaced by Liam Dick.
Falkirk grabbed the opening goal a minute before half-time when Lyle Taylor and Weatherston combined before Alston turned home Thomas Grant's cut-back.
Sean Higgins replaced Grant at half-time, but the change did little to change the momentum of the match and Fisher blasted a shot just over the top as the hosts looked lively in the early stages of the second half.
Hamilton had an incredible escape when Taylor fizzed an effort off the underside of the crossbar and then Higgins saw his goal-bound strike from the rebound kept out by Cuthbert.
Ali Crawford was introduced in place of Alex Neil just after the hour mark and the 21-year-old almost made an immediate impact - swinging over a corner that Michael Devlin headed just over the top.
The visitors continued to demonstrate an attacking threat on the counter-attack and Routledge made an excellent last-ditch challenge to prevent Taylor extending their advantage.
Hamilton were punished just 30 seconds later, however; Weatherston's corner was cleared only as far as Alston who thundered a volley beyond the helpless Cuthbert.
Andy Ryan came on for Grant Gillespie following the concession of the second goal.
It seemed that Accies' chance of a semi-final place was all but over, although they pulled themselves right back into the tie as Ryan converted after Fisher's shot was parried by McGovern.
Ryan spurned a glorious opportunity to get Hamilton back on terms when he somehow fired over after great work by May.
Reid made his final change of the match as Stephen Hendrie was introduced in place of Louis Longridge.
As Hamilton threw men forward in the closing minutes they left themselves vulnerable on the counter-attack and paid the price as Page received his marching orders for a reckless challenge on Taylor in injury-time.
- Sports & Recreation
- Kevin Cuthbert
- Gary Fisher