With relegation rivals St Mirren and Kilmarnock playing each other in Paisley, the Bairns had the chance to climb off the basement with a win.
However, not for the first time this season, Steven Pressley's side came up short despite enjoying almost total domination in the second half.
There was 19 minutes on the clock when the Portuguese midfielder found himself with all the time in the world to steer Dougie Imrie's cross past Robert Olejnik and the Lanarkshire side were good value for their interval lead.
However, after the break it was one-way traffic towards the Accies goal but the equaliser failed to arrive leaving the Bairns at the bottom of the table.
It was a classic game of two halves.
Revitalised Accies had lost once in their previous nine games but their chances appeared to be diminished in the absence of the influential midfielder James McArthur and keeper Tomas Cerny.
However, the visitors had the best of the first half and perhaps should have had the points tied up by the interval.
After only a minute Marco Paixao's corner from the left sped across the Bairns' six-yard box and needed only a touch but instead the ball was allowed to go to safety.
Seconds later at the other end, Pedro Moutinho had a right-footed shot from 16 yards which Accies' stand-in keeper Sean Murdoch saved.
The match quickly settled down in to a scrappy affair. In the 13th minute Darren Barr and Accies midfielder Simon Mensing had a spat after the Falkirk skipper went to ground inside the box when the two players clashed but referee Calum Murray nipped it in the bud. However, in the 19th minute the visitors took the lead in simple fashion.
Imrie's cross in to the box found Marco Paixao unmarked and he headed past Olejnik from 10 yards for his sixth goal of the season.
Two minutes later, in an Accies breakaway, with the Falkirk defence missing, Mickael Antoine-Curier set up Marco's twin brother Flavio Paixao but Olejnik rushed from his goal to make a crucial block.
Moments later, Bairns defender Marc Twaddle limped off injured to be replaced by Thomas Scobbie.
But Falkirk were almost handed a fortuitous equaliser when Mark Stewart's header from Barr's cross came off Trent McLenahan and bounced just past the post with Murdoch beaten.
The home side tried to battle back level but Hamilton, looking more relaxed in their play, held firm while looking dangerous on the break.
Falkirk substitute striker Enoch Showunmi replaced midfielder Burton O'Brien at the interval and with his height and power he immediately set about upsetting the Accies defence. As the Bairns swarmed around the Hamilton box Moutinho had a powerful drive pushed over the bar by Murdoch and the midfielder's in-swinging corner from the left was missed by everyone before going out for a goal-kick.
The reinvigorated home side continued on the ascendancy but could not quite fashion a clear-cut chance.
In the 56th minute Accies defender Mark McLaughlin was replaced by David Elebert before Antoine-Curier flashed a free-kick from 25 yards over the bar, after Bairns defender Kieran Duffie had been booked for a foul on James Wesolowski.
Duffie was replaced by Jack Compton in the 63rd minute as Falkirk tried to increase the pressure.
Hamilton responded by bringing on Kyle Wilkie for Martin Canning and reshuffled their defence again but as they struggled for the calm of their first-half performance, Stewart drove just over Murdoch's bar.
Accies enjoyed some luck in the 70th minute when Moutinho's corner from the right was met by Barr whose header crashed off the post, rebounded off the unwitting Antoine-Curier and back in the arms of Murdoch.
The visitors were now in retreat mode and they brought on Joel Thomas for Antoine-Curier in a bid to take the game to Falkirk.
But the home side remained in control and with four minutes remaining Moutinho had an effort from distance which went just wide, and the home side passed up on several chances to level in a dramatic finale, a familiar tale this season.