Jason Scotland provided the perfect tonic with an early goal and the striker could have made things easier for the home side had his control not let him down in front of goal late in the first half.
Falkirk, who posed little goal threat in the first half, equalised through a well-directed header from Mark Beck after 71 minutes.
But Darian MacKinnon had Hamilton back in front three minutes later following a corner kick and Tony Andreu made sure of the points when he pounced after Michael McGovern had produced a fine save from James Keatings. The result leaves Falkirk five points adrift of the front two.
Dundee kept ahead of Hamilton on goal difference with a 2-0 victory at Livingston. Peter MacDonald struck his 18th goal of the season with a 20-yard free-kick after Simon Mensing had committed a needless foul.
The Dark Blues doubled their lead when Jim McAlister found the head of Kyle Benedictus with a probing corner and the defender headed powerfully past Darren Jamieson.
Dumbarton enjoyed the best win of the day when they beat Cowdenbeath 5-1 at Bet Butler Stadium.
Mitch Megginson put the hosts ahead when he curled the ball home from the edge of the area after Colin Nish had set him up with a neat back-heel.
Michael Miller put Dumbarton two goals ahead with a low shot, but John Armstrong pulled a goal back following a corner.
Jordan Kirkpatrick made it 3-1 with a free-kick which Thomas Flynn failed to deal with and substitute Garry Fleming added the fourth from Kirkpatrick's cutback.
Chris Kane completed the scoring with a cool finish, taking the ball around Flynn. Queen of the South handed Alloa their seventh successive home defeat with a 1-0 victory at Recreation Park.
Andy Dowie headed the only goal from a Chris Mitchell free-kick after 18 minutes and Alloa could not salvage the game.
Raith Rovers ended their 10-game winless run by beating Morton 2-1 with Greig Spence contributing both goals.
Morton led through Rowan Vine, but Spence equalised from the penalty spot midway through the second half after a foul by Marc Fitzpatrick on Joe Cardle and the former Celtic frontman fired home the winner 10 minutes from time.