Top scorer James Keatings was injured in the warm-up and had to be replaced by Jon McShane but after that things pretty much went Hamilton's way as they won 2-0.
In a dominant first-half performance Michael Devlin sent an early header wide from a corner when he should have netted and Ali Crawford was denied by an agile save from Michael McGovern.
Hamilton took a deserved lead when Tony Andreu netted from close range midway through the first half from Mickael Antoine-Curier's firmly-delivered cross.
Falkirk fought back towards half-time and Kevin Cuthbert produced two decent saves to deny Rory Loy and then Craig Sibbald.
Falkirk improved in the second half but Hamilton were well organised and killed off the game three minutes from time when substitute Louis Longridge scored from the edge of the area.
Dundee saw their challenge slip when they lost 2-1 at Livingston after a penalty miss. Referee Craig Charleston awarded Dundee the spot-kick when Callum Fordyce bundled into Peter MacDonald, whose penalty was saved by Darren Jamieson.
MacDonald made amends by collecting a clearance and volleying past the stricken keeper to put Dundee in front but Livingston came back immediately, Keaghan Jacobs finding the bottom corner from the edge of the box.
Livingston's late winner came when Martin Scott netted after Coll Donaldson had been denied by Kyle Letheren. Iain Russell scored twice as Queen of the South withstood early Alloa Athletic pressure to win 3-0 at Recreation Park.
The Dumfries side took a 16th-minute lead when Kevin Holt's free-kick was touched on by Mark Durnan and Russell arrived at the right time to head home.
On-loan goalkeeper Zander Clark kept his side ahead when he leapt to save a header from Kevin Cawley. Russell scored again with 20 minutes left after he calmly rounded Scott Bain. Gavin Reilly added a third.
Dumbarton and Cowdenbeath could not find a goal between them at Bet Butler Stadium. Dumbarton's Jordan Kirkpatrick was unlucky to see a long-range effort hit the crossbar and Mitch Megginson struck the post in the second half in the 0-0 draw
Morton are five points adrift at the bottom after a cruel goal in injury-time from substitute Gordon Smith meant a 2-1 loss at Raith Rovers.
Raith took the lead with a superbly crafted goal just before half-time, Kevin Moon taking a through ball from Jason Thomson and lobbing Morton goalkeeper Nicolas Caraux.
Nacho Novo equalised for the visitors 20 minutes from time but Smith settled it.