James McCarthy had put Accies ahead 10 minutes before the break by lashing home an impressive shot following Brian Easton's throw-in, but Higdon cancelled that out with a headed equaliser.
The draw arrested Hamilton's miserable run of six straight league defeats, but the New Douglas Park side's first point since September was not enough to move them off the bottom of the table.
Hamilton manager Billy Reid made four changes to his bottom-of-the-table side.
Mark McLaughlin returned from six weeks out with a knee injury and fellow defender Chris Swailes recovered from a hamstring problem to return to the line-up alongside James McCarthy and Lucas Akins.
David Elebert, Trent McClenahan, Simon Mensing and Derek Lyle - who had all started in the 2-0 derby defeat to Motherwell seven days earlier - were on the bench.
Falkirk were unchanged from last Saturday's goalless draw against Dundee United as they sought their second away win of the season.
Hamilton carved out the first chance after four minutes when Lucas Akins showed impressive determination to keep the ball in at the right corner flag and fired in a cross which Mark Corcoran allowed to run through to McCarthy, but he fired wide at the far post.
Accies made the brighter start and forced Falkirk goalkeeper Scott Flinders into a good 13th-minute save.
He touched Akins' shot onto the base of the left-hand post after Corcoran had crossed from the left and Richard Offiong turned cleverly to set up his strike partner for a shot which looked set to sneak into the corner of the net.
Steve Lovell, playing as a lone striker for the visitors ahead of a five-man midfield, had Falkirk's first effort of note when he flashed a diagonal effort wide after 19 minutes.
Hamilton left-back Easton scampered off on an excellent run from the halfway line on 25 minutes, streaking through the visiting defence and crossing for the well-positoned Akins at the far post.
But the striker got the ball caught under his feet and contrived to turn the chance wide.
Burton O'Brien had a half-chance for Falkirk just after the half-hour but his shot from the edge of the area rebounded off the Hamilton defence.
However, Accies finally broke the deadlock 10 minutes from the interval.
Easton's throw from the left bobbled around the penalty area and with the visitors unable to clear their lines, the ball fell to midfielder McCarthy, who lashed home an unstoppable right-foot shot.
Falkirk responded with Darren Barr heading over the top of Tomas Cerny's goal, while opposition captain Alex Neil curled a stoppage-time free kick over the bar.
Visiting manager John Hughes responded at half-time by replacing midfielder Patrick Cregg with forward Higdon and reshaping his side to 4-4-2.
It paid immediate dividends as the substitute equalised within two minutes, heading home an O'Brien cross which was flicked on to him at the opposite post.
O'Brien then tried his luck with a 30-yard effort which flew over, while a bizarre moment just before the hour saw Flinders' attempted clearance cannon back off McLaughlin and drift only narrowly past the left-hand post.
However, Hamilton continued attacking and were only denied a second goal when Flinders stretched to tip a powerful McCarthy effort over the bar.
Barr was disappointed to pass up a good chance to put Falkirk in front for the first time with a free header following Neil McCann's corner from the left, driving his attempt down into the turf and seeing it bounce off target.
The game became somewhat heated in the final quarter as both teams searched for the winning goal, with O'Brien and Arfield to the fore for Falkirk.
Joel Thomas stretched for a cross from fellow Hamilton substitute Lyle, but could only touch the ball into Flinders' reaching arms.
McCarthy launched a cross from the right 15 minutes from time which fizzed dangerously through the box, but Flinders was first to react and gathered it at the far post.
Gerard Aafjes put Neil McCann's cross past the right-hand post for Falkirk. And, at the other end, David Graham came just as close from Easton's cross and McLaughlin's connection with a long ball from Neil was also just off target.
Flinders then kept out Thomas as he tried to weave along the byline in the closing minutes, and opposite number Cerny blocked Arfield in the last minute.