Accies keeper Tomas Cerny was the busier goalkeeper with his first-half block from Kjartan Finnbogason's volley the pick of his saves.
However, Finnigan did beat him with a close-range header seven minutes from the break, only to see it ruled out by referee Willie Collum for a foul on Accies defender Martin Canning.
Falkirk had the best of the first half but they fell away after the break, leaving them no more deserving of a point than Accies.
Hamilton boss Billy Reid's plans suffered an early blow when, with less than a minute gone, striker Derek Lyle collapsed after his studs stuck in the turf as he tried to beaver his way in to the Bairns box.
The reaction of players from both sides suggested a serious injury.
The game stopped for around six minutes until Lyle was settled by the paramedics, then the former Dundee striker was stretchered from the field to be replaced by David Louhoungou.
The match took a few moments to get going again and in the 10th minute Finnbogason had a decent effort from distance which just evaded the bar.
Four minutes later, after Falkirk left-back Thomas Scobbie had fouled Simon Mensing 25 yards from goal, Accies defender Richard Hastings drew a fine save from Robert Olejnik with a curling free-kick.
Finnbogason came closer when Sean Lynch found some space on the right-hand side of the box to cut the ball back to the Icelandic striker, whose first-time volley from 16 yards was brilliantly parried away by diving Cerny.
On the half-hour mark Falkirk midfielder Ryan Flynn tried his luck with 30-yards drive but it deflected kindly in to the arms of Cerny.
Accies were playing second fiddle but they carved out a half-chance when Hastings threw a searching cross in to the Bairns box from the left but Louhoungou looped his header from 12 yards over the bar.
There was a moment of controversy in the 38th minute when Finnigan had the ball in the Accies net from a Lynch cross.
However, referee Collum adjudged the Bairns' striker to have fouled Canning in the process of heading past Cerny from around six yards out. The home side, disappointing in the first 45 minutes, looked no more coherent after the break.
Falkirk might have taken the lead in the 50th minute when Danijel Marceta drove a left-footed cross in from the right to find Finnbogason racing in on goal.
However, the Bairns' striker failed to judge the speed of delivery and the ball came off his body six yards from goal before going past the post.
In the 57th minute Accies almost stole in to the lead when Louhoungou's pass down the left channel fortuitously landed at the feet of Mickael Antoine-Curier via a deflection off Jackie McNamara and Brian McLean's slip.
The big striker drove in to the box and unleashed a drive but the ball took a nick off Falkirk stopper Darren Barr and arced over Olejnik and on to the roof of the net, with the visitors surviving the corner.
Falkirk had lost their dominance from the first half and as the match moved in to the last quarter, Accies began to offer the occasional threat.
But the game was declining in quality, with both sets of players lacking guile and imagination.
Even the traditional late rally for the winner failed to materialise - aside from a drive from Louhoungou which went wide - and in the end, Falkirk picked up a point but remain bottom of the table.